Understanding exponents - GMAT Quantitative
Card 0 of 672
Which of the following numbers is in scientific notation?
Which of the following numbers is in scientific notation?
The number in front must have an absolute value greater than or equal to 1, and less than 10. Of these choices, only
qualifies.
The number in front must have an absolute value greater than or equal to 1, and less than 10. Of these choices, only qualifies.
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Putting these together,

Putting these together,
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Then, 
Then,
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Then putting them together, 
Then putting them together,
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If
, what is
in terms of
?
If , what is
in terms of
?
We have
.
So
, and
.
We have .
So , and
.
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Which of the following expressions is equivalent to this expression?

You may assume that
.
Which of the following expressions is equivalent to this expression?
You may assume that .
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Simplify the following expression without a calculator:

Simplify the following expression without a calculator:
The easiest way to simplify is to work from the inside out. We should first get rid of the negatives in the exponents. Remember that variables with negative exponents are equal to the inverse of the expression with the opposite sign. For example,
So using this, we simplify: 
Now when we multiply variables with exponents, to combine them, we add the exponents together. For example, 
Doing this to our expression we get it simplified to
.
The next step is taking the inside expression and exponentiating it. When taking an exponent of a variable with an exponent, we actually multiply the exponents. For example,
. The other rule we must know that is an exponent of one half is the same as taking the square root. So for the
So using these rules, 
The easiest way to simplify is to work from the inside out. We should first get rid of the negatives in the exponents. Remember that variables with negative exponents are equal to the inverse of the expression with the opposite sign. For example, So using this, we simplify:
Now when we multiply variables with exponents, to combine them, we add the exponents together. For example,
Doing this to our expression we get it simplified to .
The next step is taking the inside expression and exponentiating it. When taking an exponent of a variable with an exponent, we actually multiply the exponents. For example, . The other rule we must know that is an exponent of one half is the same as taking the square root. So for the
So using these rules,
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If
, what does
equal?
If , what does
equal?
We can use the fact that
to see that 
Since
, we have
.
We can use the fact that to see that
Since , we have
.
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Rewrite as a single logarithmic expression:

Rewrite as a single logarithmic expression:
First, write each expression as a base 3 logarithm:
since 

Rewrite the expression accordingly, and apply the logarithm sum and difference rules:




First, write each expression as a base 3 logarithm:
since
Rewrite the expression accordingly, and apply the logarithm sum and difference rules:
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What are the last two digits, in order, of
?
What are the last two digits, in order, of ?
Inspect the first few powers of 6; a pattern emerges.










As you can see, the last two digits repeat in a cycle of 5.
789 divided by 5 yields a remainder of 4; the pattern that becomes apparent in the above list is that if the exponent divided by 5 yields a remainder of 4, then the power ends in the diigts 96.
Inspect the first few powers of 6; a pattern emerges.
As you can see, the last two digits repeat in a cycle of 5.
789 divided by 5 yields a remainder of 4; the pattern that becomes apparent in the above list is that if the exponent divided by 5 yields a remainder of 4, then the power ends in the diigts 96.
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Which of the following expressions is equal to the expression
?
Which of the following expressions is equal to the expression
?
Use the properties of exponents as follows:





Use the properties of exponents as follows:
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Simplify:

Simplify:
Use the properties of exponents as follows:






Use the properties of exponents as follows:
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Simplify:
![\left [ \left ( x ^{5} \right )^{4} \right ]^{3}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/170965/gif.latex)
Simplify:
Apply the power of a power principle twice by multiplying exponents:
![\left [ \left ( x ^{5} \right )^{4} \right ]^{3} = \left ( x ^{5 \cdot 4} \right ) ^{3} = \left ( x ^{20} \right ) ^{3} = x ^{20 \cdot 3} = x ^{60 }](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/170966/gif.latex)
Apply the power of a power principle twice by multiplying exponents:
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Divide:

Divide:
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Solve for
:

Solve for :



The left and right sides of the equation have the same base, so we can equate the exponents and solve:





The left and right sides of the equation have the same base, so we can equate the exponents and solve:
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Solve for
:

Solve for :
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Which of the following is equal to
?
Which of the following is equal to ?
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Which of the following is equal to
?
Which of the following is equal to ?
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Which of the following is true about
?
Which of the following is true about ?

Therefore,



Therefore,
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Which of the following is equal to
?
Which of the following is equal to ?
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