Exponents & Roots - ACT Math
Card 0 of 561
Whenever you exponential expressions in both the numerator and denominator of a fraction, your first inclination might be to quickly simplify the expressions by canceling terms out in the numerator and denominator.
However, remember to follow the order of operations: you must simplify the numerator and denominator separately to revolve the exponents raised to exponents problem before you can look to cancel terms in the numerator and denominator.
If you consider the numerator, 
, you should recognize that, because there is no addition or subtraction within the parentheses, that you can simplify this expression by multiplying each exponent within the parentheses by 
to get:
.
Similarly, you can simplify the denominator, 
by multiplying each exponent within the parentheses by 
to get:
.
You can then recombine the numerator and denominator to get 
. Notice that you now have simple division. Remember that anytime you want to combine two exponential expressions with a common base that are being divided, you simply need to subtract the exponents. If you do, you get:
, which simplifies to 
.
Remember that if you forgot any of the rules, such as how to combine exponents with common bases that are being divided or whether you add or multiply when raising a power to a power, that you can always remind yourself how the rules work by testing small numbers.
Whenever you exponential expressions in both the numerator and denominator of a fraction, your first inclination might be to quickly simplify the expressions by canceling terms out in the numerator and denominator.
However, remember to follow the order of operations: you must simplify the numerator and denominator separately to revolve the exponents raised to exponents problem before you can look to cancel terms in the numerator and denominator.
If you consider the numerator,
Similarly, you can simplify the denominator,
You can then recombine the numerator and denominator to get
Remember that if you forgot any of the rules, such as how to combine exponents with common bases that are being divided or whether you add or multiply when raising a power to a power, that you can always remind yourself how the rules work by testing small numbers.
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Whenever you are asked to simplify an expression with exponents and two different bases, you should immediately look to factor. In this case, you should notice that both 
and 
are powers of 
. This means that you can rewrite them as 
and 
respectively.
Once you do this, the numerator becomes 
.
As you are raising an exponent to an exponent, you should then recognize that you need to multiply the two exponents in order to simplify to get 
.
Similarly, the denominator becomes 
, which you can simplify by multiplying the exponents to get 
.
Your fraction is therefore 
. Remember that to divide exponents of the same base, simply subtract the exponents. This gives you:
, or 
.
Whenever you are asked to simplify an expression with exponents and two different bases, you should immediately look to factor. In this case, you should notice that both
Once you do this, the numerator becomes
As you are raising an exponent to an exponent, you should then recognize that you need to multiply the two exponents in order to simplify to get
Similarly, the denominator becomes
Your fraction is therefore
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What is 
?
What is
Whenever you see addition or subtraction with algebraic terms, you should only think about combining like terms or factoring. Here you have two of one term 
and three of another term 
so:
The difficulty in this problem relates primarily to common mistakes with factoring and exponent rules. If you understand exponent rules and how to combine like terms, you will answer this problem quickly and confidently. You should note that the answer choices do not really help you here – they are traps if you make a mistake with exponent rules! Many algebra problems on the ACT exploit common mistakes relating to certain content areas. For instance, in this example, you can see how easy it would be to accidentally add the exponents or add the bases. If you ever make one of these common mistakes, take note and be sure to avoid it the next time you see a similar problem.
Whenever you see addition or subtraction with algebraic terms, you should only think about combining like terms or factoring. Here you have two of one term
The difficulty in this problem relates primarily to common mistakes with factoring and exponent rules. If you understand exponent rules and how to combine like terms, you will answer this problem quickly and confidently. You should note that the answer choices do not really help you here – they are traps if you make a mistake with exponent rules! Many algebra problems on the ACT exploit common mistakes relating to certain content areas. For instance, in this example, you can see how easy it would be to accidentally add the exponents or add the bases. If you ever make one of these common mistakes, take note and be sure to avoid it the next time you see a similar problem.
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can be rewritten as:
This problem tests your ability to combine exponents algebraically, using both the distributive property and the rule for multiplying exponents with the same base. Here it is also helpful to look at the answers to see what the test maker is looking for. In the answer choices, the maximum number of individual terms is 2, and all terms involve a base 10 (or 100). So you should see that your goal is to rewrite as much as possible of what you're given in terms of 10.
In terms of factors/multiples, a 10 is created any time you can pair a 2 with a 5. So as you take the given expression:
Recognize that if you can pair the two 2s you have with two of the 5s in 5555, you'll be able to consider them 10s. So you can rewrite the given expression as:
From there, you can combine the 
and 
terms:
And since those are two bases, multiplied, each taken to the same exponent, they'll combine to
. That can be rewritten as 
, making your expression:
From here, you'll apply the rule that 
and add the exponents from the 10s. That gives you:
This problem tests your ability to combine exponents algebraically, using both the distributive property and the rule for multiplying exponents with the same base. Here it is also helpful to look at the answers to see what the test maker is looking for. In the answer choices, the maximum number of individual terms is 2, and all terms involve a base 10 (or 100). So you should see that your goal is to rewrite as much as possible of what you're given in terms of 10.
In terms of factors/multiples, a 10 is created any time you can pair a 2 with a 5. So as you take the given expression:
Recognize that if you can pair the two 2s you have with two of the 5s in 5555, you'll be able to consider them 10s. So you can rewrite the given expression as:
From there, you can combine the
And since those are two bases, multiplied, each taken to the same exponent, they'll combine to
From here, you'll apply the rule that
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What is 
?
What is
The key to beginning this problem is finding common bases. Since 2, 4, and 8 can all be expressed as powers of 2, you will want to factor the 4 into 
and the 8 into 
so that 
can be rewritten as 
.
From there, you will employ two core exponent rules. First, when you take an exponent to another, you'll multiply the exponents.
That means that:
becomes 
and
becomes 
So your new expression is 
.
Then, when you're multiplying exponents of the same base, you add the exponents. So you can sum 2 + 8 + 24 to get 34, making your simplified exponent 
.
The key to beginning this problem is finding common bases. Since 2, 4, and 8 can all be expressed as powers of 2, you will want to factor the 4 into
From there, you will employ two core exponent rules. First, when you take an exponent to another, you'll multiply the exponents.
That means that:
and
So your new expression is
Then, when you're multiplying exponents of the same base, you add the exponents. So you can sum 2 + 8 + 24 to get 34, making your simplified exponent
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With this exponent problem, the key to getting the given expression in actionable form is to find common bases. Since both 9 and 27 are powers of 3, you can rewrite the given expression as:
When you've done that, you're ready to apply core exponent rules. When you take one exponent to another, you multiply the exponents. So for your numerator:
becomes 
. And for your denominator: 
becomes 
. So your new fraction is: 
Next deal with the negative exponents, which means that you'll flip each term over the fraction bar and make the exponent positive. This then makes your fraction:
From there, recognize that when you divide exponents of the same base, you subtract the exponents. This means that you have:
which simplifies to 
, so your answer is 
.
With this exponent problem, the key to getting the given expression in actionable form is to find common bases. Since both 9 and 27 are powers of 3, you can rewrite the given expression as:
When you've done that, you're ready to apply core exponent rules. When you take one exponent to another, you multiply the exponents. So for your numerator:
Next deal with the negative exponents, which means that you'll flip each term over the fraction bar and make the exponent positive. This then makes your fraction:
From there, recognize that when you divide exponents of the same base, you subtract the exponents. This means that you have:
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What is 
?
What is
This problem rewards those who see that roots and exponents are the same operations (roots are "fractional exponents"), and who therefore choose the easier order in which to perform the calculation. The trap here is to have you try to square 27. Not only is that labor-intensive, but once you get to 729 you then have to figure out how to take the cube root of that!
Because you can handle the root and the exponent in either order (were you to express this as a fractional exponent, it would be 
, which proves that the root and exponent are the same operations), you can take the cube root of 27 first if you want to, which you should know is 3. So at that point, your problem is what is 
?" And you, of course, know the answer: it's 9.
This problem rewards those who see that roots and exponents are the same operations (roots are "fractional exponents"), and who therefore choose the easier order in which to perform the calculation. The trap here is to have you try to square 27. Not only is that labor-intensive, but once you get to 729 you then have to figure out how to take the cube root of that!
Because you can handle the root and the exponent in either order (were you to express this as a fractional exponent, it would be
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can be expressed as:
It is important to be able to convert between root notation and exponent notation. The third root of a number (for example, 
is the same thing as taking that number to the one-third power 
.
So when you see that you're taking the third root of 
, you can read that as 
to the 
power:
This then allows you to apply the rule that when you take one exponent to another power, you multiply the powers:
This then means that you can express this as:
It is important to be able to convert between root notation and exponent notation. The third root of a number (for example,
So when you see that you're taking the third root of
This then allows you to apply the rule that when you take one exponent to another power, you multiply the powers:
This then means that you can express this as:
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can be expressed as:
With roots, it is important that you are comfortable with factoring and with expressing roots as fractional exponents. A square root, for example, can be expressed as taking that base to the 
power. Using that rule, the given expression, 
, could be expressed using fractional exponents as:
This would allow you to then add the exponents and arrive at:
Since that 2 in the denominator of the exponent translates to "square root," you would have the square root of 
:
If you were, instead, to work backward from the answer choices, you would see that answer choice 
factors to the given expression. If you start with:
You can express that as:
That in turn will factor to:
The first root then simplifies to 
, leaving you with:
Therefore, as you can see, choice 
factors directly back to the given expression.
With roots, it is important that you are comfortable with factoring and with expressing roots as fractional exponents. A square root, for example, can be expressed as taking that base to the
This would allow you to then add the exponents and arrive at:
Since that 2 in the denominator of the exponent translates to "square root," you would have the square root of
If you were, instead, to work backward from the answer choices, you would see that answer choice
You can express that as:
That in turn will factor to:
The first root then simplifies to
Therefore, as you can see, choice
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Which of the following is equal to 
for all positive values of 
?
Which of the following is equal to
Simplify each of the expressions to determine which satisfies the condition of the problem:
Simplify each of the expressions to determine which satisfies the condition of the problem:
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Simplify:
Simplify:
It can be easier to line real and imaginary parts vertically to keep things organized, but in essence, combine like terms (where 'like' here means real or imaginary):
It can be easier to line real and imaginary parts vertically to keep things organized, but in essence, combine like terms (where 'like' here means real or imaginary):
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Which of the following complex numbers is equal to 
? (Note: 
)
Which of the following complex numbers is equal to
When adding and subtracting complex numbers, the “
” functions just like a regular variable, the same as if it were “
” or any other letter variable. It is only when multiplying and dividing complex numbers that there is a special step where 
is transformed into 
. This question simply asks you to subtract one complex number from another one, so “
” functions just like any other letter variable.
First subtract the real number parts: 
. Then subtract the imaginary number parts: 
. Putting the parts together gives you the correct answer choice 
.
The tricky step is 
. Subtracting a negative is the same thing as adding a positive since the negative signs cancel each other out. Failure to do this step correctly can lead to the wrong answer choice 
.
Students who are expecting the question to be more challenging than it actually is may see the complex numbers in parentheses and jump to the conclusion that they need to multiply the complex numbers rather than just subtract them. Incorrectly multiplying the complex numbers 
rather than subtracting them will lead to the wrong answer choice 
.
Multiplying the complex numbers and also making a mistake when multiplying 
may lead to the wrong answer choice 
.
When adding and subtracting complex numbers, the “
First subtract the real number parts:
The tricky step is
Students who are expecting the question to be more challenging than it actually is may see the complex numbers in parentheses and jump to the conclusion that they need to multiply the complex numbers rather than just subtract them. Incorrectly multiplying the complex numbers
Multiplying the complex numbers and also making a mistake when multiplying
Compare your answer with the correct one above
Which of the following complex numbers is equal to 
, where 
?
Which of the following complex numbers is equal to
When adding and subtracting complex numbers, the “
” functions just like a regular variable, the same as if it were “
” or any other letter variable. It is only when multiplying and dividing complex numbers that there is the special step where 
is transformed into 
. This question simply asks you to subtract one complex number from another one, so “
” functions just like any other letter variable.
First subtract the real number parts: 
. Then subtract the imaginary number parts: 
. Putting the parts together gives you the correct answer choice 
.
To do both these subtraction steps correctly, the student needs to keep track of all the signs accurately and understand that subtracting a negative is the same thing as adding a positive, since the negative signs cancel each other out. Failure to do this step correctly with the imaginary number parts can lead to the wrong answer choice 
. Failure to do this step correctly with the real number parts and the imaginary number parts can lead to the wrong answer choice 
.
Students who are expecting the question to be more challenging than it actually is may see the complex numbers in parentheses and jump to the conclusion that they need to multiply the complex numbers rather than just subtract them. Incorrectly multiplying the complex numbers 
rather than subtracting them will lead to the wrong answer choice 
.
When adding and subtracting complex numbers, the “
First subtract the real number parts:
To do both these subtraction steps correctly, the student needs to keep track of all the signs accurately and understand that subtracting a negative is the same thing as adding a positive, since the negative signs cancel each other out. Failure to do this step correctly with the imaginary number parts can lead to the wrong answer choice
Students who are expecting the question to be more challenging than it actually is may see the complex numbers in parentheses and jump to the conclusion that they need to multiply the complex numbers rather than just subtract them. Incorrectly multiplying the complex numbers
Compare your answer with the correct one above
What is the sum 
? (Note: 
)
What is the sum
When adding and subtracting complex numbers, the “
” functions just like a regular variable, the same as if it were “
” or any other letter variable. It is only when multiplying and dividing complex numbers that there is the special step where 
is transformed into 
. This question simply asks you to add two complex numbers, so “
” functions just like any other letter variable.
First add the real number parts 
. Then add the imaginary number parts 
. Putting the parts together gives you the correct answer choice 
.
It is easy to get confused when adding two negative numbers, which is the other main challenge in this question. Many students see it as a subtraction operation, so they want to say that 
. But when both numbers are negative, it works more like addition, but with every number having a negative sign. Failure to understand this concept and perform this operation correctly with the real number parts may lead to the wrong answer choice 
. Failure to understand this concept and perform this operation correctly with the imaginary number parts may lead to the wrong answer choice 
Failure to understand this concept and perform this operation correctly with both parts may lead to the wrong answer choice 
.
When adding and subtracting complex numbers, the “
First add the real number parts
It is easy to get confused when adding two negative numbers, which is the other main challenge in this question. Many students see it as a subtraction operation, so they want to say that
Compare your answer with the correct one above
Which of the following complex numbers is equal to 
, for 
?
Which of the following complex numbers is equal to
There are a few tricks to this question. First of all, you must carefully observe the minus sign in the middle of the expression, which means that you are subtracting the complex numbers, not multiplying them!
Second, you must note the term 
in the second complex number. Now you have to know that because 
, therefore 
. So 
. Now our whole expression is 
.
Finally, now you have to be very careful about all the minus signs! For both of the terms in the second complex number, you are now subtracting a negative, which means that you are actually just doing the same thing as adding a positive! Therefore, the whole expression is simply equivalent to 
. Now it looks much easier, doesn’t it? Simply adding together the real number parts and the imaginary number parts separately, you get the final answer, the correct answer choice 
.
If you do not simplify 
correctly, or if you accidentally subtract 
instead of add 
in the final step, you may get one of the wrong answer choices 
or 
. If you accidentally subtract 
instead of add 
in the final step, you may get one of the wrong answer choices 
or 
.
There are a few tricks to this question. First of all, you must carefully observe the minus sign in the middle of the expression, which means that you are subtracting the complex numbers, not multiplying them!
Second, you must note the term
Finally, now you have to be very careful about all the minus signs! For both of the terms in the second complex number, you are now subtracting a negative, which means that you are actually just doing the same thing as adding a positive! Therefore, the whole expression is simply equivalent to
If you do not simplify
Compare your answer with the correct one above
Which of the following complex numbers is equal to 
, for 
?
Which of the following complex numbers is equal to
There are a few tricks to this question. First of all, you must carefully observe the minus sign in the middle of the expression, which means that you are subtracting the complex numbers, not multiplying them!
Second, you must note the term 
in the second complex number. Now you have to know that because 
, therefore 
. So 
. Now our whole expression is 
.
Finally, now you have to be very careful about all the minus signs! For both of the terms in the second complex number, you are now subtracting a negative, which means that you are actually just doing the same thing as adding a positive! Therefore, the whole expression is simply equivalent to 
. Now it looks much easier, doesn’t it? Simply adding together the real number parts and the imaginary number parts separately, you get the final answer, the correct answer choice 
.
If you do not simplify 
correctly, or if you accidentally subtract 
instead of add 
in the final step, you may mistakenly think the “
” and “
” terms cancel out, and get one of the wrong answer choices 
or 
. If you accidentally subtract 
instead of add 
in the final step, you may get one of the wrong answer choices 8i or 
.
There are a few tricks to this question. First of all, you must carefully observe the minus sign in the middle of the expression, which means that you are subtracting the complex numbers, not multiplying them!
Second, you must note the term
Finally, now you have to be very careful about all the minus signs! For both of the terms in the second complex number, you are now subtracting a negative, which means that you are actually just doing the same thing as adding a positive! Therefore, the whole expression is simply equivalent to
If you do not simplify
Compare your answer with the correct one above
Which of the following complex numbers is equivalent to 
? (Note: 
)
Which of the following complex numbers is equivalent to
This is the classic more difficult type of complex number/imaginary number question on the ACT math section: Division of one complex number by another complex number. The most difficult and critical key step of the solution process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra: 
. This formula works because when you FOIL 
, the two middle terms 
will cancel each other out. Well, the same thing happens here when you multiply the complex number in the denominator by its complex conjugate: the imaginary number middle terms will cancel each other out! So 
. The middle terms 
cancel each other out, leaving us with 
. Now you have to know that because 
, therefore 
So 
, and now the negative signs in the second term cancel out, making it positive: 
. The point of this whole process is that the end result is to simplify the denominator to a single real number, which in this case is 
.
To finish solving the question, now you must also multiply the original numerator by the complex conjugate of the denominator: 
. The two middle terms, the imaginary parts, combine: 
.
Pro Tip: If you are alert here, you can already see that the imaginary part of the final answer will have 
in the numerator and 
in the denominator! This alone eliminates all of the wrong answer choices, so you can already see now that the correct answer choice must be 
!
If you do need to finish the solution and know the numerator of the real part as well, you continue 
, since you must know that 
as we stated above. Finishing the simplification of the numerator, we get 
. Now we see how the entire correct answer choice 
is right.
You may wonder how experienced math students can finish solving questions so fast and complete the ACT math section within the strict time limits and still get perfect ACT Math scores. The answer is, they use the four answer choices to guide them and speed up the process dramatically. Look at the answer choices 
and 
. Do you see how they have exactly the same numbers and coefficients as the numerator and denominator of the expression in the original question? Well, experienced math students know that simplification of a fraction like this one with two or more terms in the denominator that are added or subtracted is never this simple! It always involved a more complicated process that inevitably changes the numbers and coefficients in the final answer. So experienced math students can eliminate these two answer choices instantly as soon as they look at them.
Then, looking at the other two answer choices 
and 
, experienced math students see that the only difference between them is the sign in the middle, before the imaginary part. So the only step they have to do is to multiply the numerator by the complex conjugate of the denominator, 
. As soon as they see that the imaginary part of the numerator will have to be negative, 
, they already instantly know that 
must be wrong, and 
must be right. (See the Pro Tip above.) This is how the top ACT math students are able to answer difficult questions like this one so quickly and correctly.
This is the classic more difficult type of complex number/imaginary number question on the ACT math section: Division of one complex number by another complex number. The most difficult and critical key step of the solution process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra:
To finish solving the question, now you must also multiply the original numerator by the complex conjugate of the denominator:
Pro Tip: If you are alert here, you can already see that the imaginary part of the final answer will have
If you do need to finish the solution and know the numerator of the real part as well, you continue
You may wonder how experienced math students can finish solving questions so fast and complete the ACT math section within the strict time limits and still get perfect ACT Math scores. The answer is, they use the four answer choices to guide them and speed up the process dramatically. Look at the answer choices
Then, looking at the other two answer choices
Compare your answer with the correct one above
Which of the following complex numbers is equivalent to 
? (Note: 
)
Which of the following complex numbers is equivalent to
This is the classic more difficult type of complex number/imaginary number question on the ACT math section: Division of one complex number by another complex number. The most difficult and critical key step of the solution process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra: . This formula works because when you FOIL , the two middle terms will cancel each other out. Well, the same thing happens here when you multiply the complex number in the denominator by its complex conjugate: the imaginary number middle terms will cancel each other out! So 
. The middle terms 
cancel each other out, leaving us with 
. Now you have to know that because , therefore So 
, and now the negative signs in the second term cancel out, making it positive: 
. The point of this whole process is that the end result is to simplify the denominator to a single real number, which in this case is 
.
To finish solving the question, now you must also multiply the original numerator by the complex conjugate of the denominator: 
. The two middle terms, the imaginary parts, combine: 
.
Pro Tip: If you are alert here, you can already see that the imaginary part of the final answer will have 
in the numerator and 
in the denominator! This alone eliminates all of the wrong answer choices, so you can already see now that the correct answer choice must be 
!
If you do need to finish the solution and know the numerator of the real part as well, you continue 
, since you must know that as we stated above. Finishing the simplification of the numerator, we get 
. Now we see how the entire correct answer choice 
is right.
You may wonder how experienced math students can finish solving questions so fast and complete the ACT math section within the strict time limits and still get perfect ACT Math scores. The answer is, they use the four answer choices to guide them and speed up the process dramatically. Look at the answer choices 
and 
. Do you see how they have exactly the same numbers and coefficients as the numerator and denominator of the expression in the original question? Well, experienced math students know that simplification of a fraction like this one with two or more terms in the denominator that are added or subtracted is never this simple! It always involved a more complicated process that inevitably changes the numbers and coefficients in the final answer. So experienced math students can eliminate these two answer choices instantly as soon as they look at them.
Then, looking at the other two answer choices 
and 
, experienced math students see that the only difference between them is the sign in the middle, before the imaginary part. So the only step they have to do is to multiply the numerator by the complex conjugate of the denominator, 
. As soon as they see that the imaginary part of the numerator will have to be negative, 
, they already instantly know that 
must be wrong, and 
must be right. (See the Pro Tip above.) This is how the top ACT math students are able to answer difficult questions like this one so quickly and correctly.
This is the classic more difficult type of complex number/imaginary number question on the ACT math section: Division of one complex number by another complex number. The most difficult and critical key step of the solution process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra:
To finish solving the question, now you must also multiply the original numerator by the complex conjugate of the denominator:
Pro Tip: If you are alert here, you can already see that the imaginary part of the final answer will have
If you do need to finish the solution and know the numerator of the real part as well, you continue
You may wonder how experienced math students can finish solving questions so fast and complete the ACT math section within the strict time limits and still get perfect ACT Math scores. The answer is, they use the four answer choices to guide them and speed up the process dramatically. Look at the answer choices
Then, looking at the other two answer choices
Compare your answer with the correct one above
If the expression 
is rewritten in the form 
, where 
and 
are real numbers, what is the value of 
? (Note: 
)
If the expression
The twist in this question is that you must be sure to answer the question they are actually asking you! The question does not ask you to provide the entire value of the complex number that you get from solving the division problem--it only asks for the real number part of that complex number, which is called 
. (Please note: Even though 
itself is also a real number, as the question states, in the expression 
this value 
is the coefficient of 
, so it is part of the imaginary number part of the expression.) Therefore, after you perform the division of complex numbers correctly, you have to focus only on the real number part of your answer, not on the imaginary number part.
But first, you do have to perform the division of complex numbers correctly. The most difficult and critical key step of this process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra: 
. This formula works because when you FOIL 
, the two middle terms 
will cancel each other out. Well, the same thing happens here when you multiply the complex number in the denominator by its complex conjugate: the imaginary number middle terms will cancel each other out! So 
. The middle terms 
cancel each other out, leaving us with 
. Now you have to know that because 
, therefore 
. So 
, and now the negative signs in the second term cancel out, making it positive: 
. The point of this whole process is that the end result is to simplify the denominator to a single real number, which in this case is 
.
To continue the complex number division process, now you must also multiply the original numerator by the complex conjugate of the denominator: 
. The two middle terms, the imaginary parts, combine: 
. Continuing, you must know that 
as we stated above, so 
. Finishing the simplification of the numerator, we get 
.
Conveniently, we now see that our numerator 
and our denominator 
(see above) simplify dramatically! 
. Wow! (Note: On the ACT, these things never happen by accident. The test question writers carefully design the questions so that they often simplify dramatically like this, when the student solves the questions correctly.)
Now, by solving the complex number division process correctly, we have rewritten the original expression in the form 
, which in this case has the value 
. Recall that you must answer the question they are actually asking you, and the question asks you for the value of 
. Therefore the correct answer choice is 
.
Pro Tip: Experienced math students may be able to save a little time during the solution process: If you know that the question is only asking for the real number part of the final complex number, then when you solve the numerator above, you can skip the imaginary number parts, which are the two middle terms when you FOIL 
. So you can just solve the terms that will become the real number parts: 
. Since this is the real number part of the numerator, and the whole denominator is also 
, now you already know the final real number part will be 
, which is what the question is actually asking for, so that is the final correct answer choice for this question.
The twist in this question is that you must be sure to answer the question they are actually asking you! The question does not ask you to provide the entire value of the complex number that you get from solving the division problem--it only asks for the real number part of that complex number, which is called
But first, you do have to perform the division of complex numbers correctly. The most difficult and critical key step of this process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra:
To continue the complex number division process, now you must also multiply the original numerator by the complex conjugate of the denominator:
Conveniently, we now see that our numerator
Now, by solving the complex number division process correctly, we have rewritten the original expression in the form
Pro Tip: Experienced math students may be able to save a little time during the solution process: If you know that the question is only asking for the real number part of the final complex number, then when you solve the numerator above, you can skip the imaginary number parts, which are the two middle terms when you FOIL
Compare your answer with the correct one above
If the expression 
is rewritten in the form 
, where 
and 
are real numbers, what is the value of 
? (Note: 
)
If the expression
The twist in this question is that you must be sure to answer the question they are actually asking you! The question does not ask you to provide the entire value of the complex number that you get from solving the division problem--it only asks for the coefficient of the imaginary number part of that complex number, which is called 
. (Please note: Even though 
itself is a real number, as the question states, in the expression 
this value b is the coefficient of 
, so it is part of the imaginary number part of the expression.) Therefore, after you perform the division of complex numbers correctly, you have to focus only on the coefficient of the imaginary number part of your answer, not on the real number part.
But first you do have to perform the division of complex numbers correctly. The most difficult and critical key step of this process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra: 
. This formula works because when you FOIL 
, the two middle terms 
will cancel each other out. Well, the same thing happens here when you multiply the complex number in the denominator by its complex conjugate: the imaginary number middle terms will cancel each other out! So 
. The middle terms 
cancel each other out, leaving us with 
. Now you have to know that because 
, therefore 
. So 
, and now the negative signs in the second term cancel out, making it positive: 
. The point of this whole process is that the end result is to simplify the denominator to a single real number, which in this case is 
. To continue the complex number division process, now you must also multiply the original numerator by the complex conjugate of the denominator: 
. The two middle terms, the imaginary parts, combine: 
. Continuing, you must know that 
as we stated above, so 
. Finishing the simplification of the numerator, we get 
.
Conveniently, we now see that our numerator 
and our denominator 
(see above) simplify dramatically! 
. Wow! (Note: On the ACT, these things never happen by accident. The test question writers carefully design the questions so that they often simplify dramatically like this, when the student solves the questions correctly.) Now, by solving the complex number division process correctly, we have rewritten the original expression in the form 
which in this case has the value 
. Recall that you must answer the question they are actually asking you, and the question asks you for the value of 
. Therefore the correct answer choice is 
. (Note: “
” is the same value as “
”, so the coefficient is 
, even though the number 
is usually not written before the “
” in this case since it is not necessary.)
Pro Tip: Experienced math students may be able to save a little time during the solution process: If you know that the question is only asking for the coefficient of the imaginary number part of the final complex number, then when you solve the numerator above, you can skip the real number parts, which are the first and last terms when you FOIL 
. So you can just solve the terms that will become the imaginary number parts: 
. Since this is the imaginary number part of the numerator, and the whole denominator is 
, now you already know the final imaginary number part will be 
, and its coefficient is 
, which is what the question is actually asking for, so that is the final correct answer choice for this question.
The twist in this question is that you must be sure to answer the question they are actually asking you! The question does not ask you to provide the entire value of the complex number that you get from solving the division problem--it only asks for the coefficient of the imaginary number part of that complex number, which is called
But first you do have to perform the division of complex numbers correctly. The most difficult and critical key step of this process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra:
Conveniently, we now see that our numerator
Pro Tip: Experienced math students may be able to save a little time during the solution process: If you know that the question is only asking for the coefficient of the imaginary number part of the final complex number, then when you solve the numerator above, you can skip the real number parts, which are the first and last terms when you FOIL
Compare your answer with the correct one above