Square Roots and Operations - PSAT Math
Card 0 of 210
The square root of 5184 is:
The square root of 5184 is:
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The easiest way to narrow down a square root from a list is to look at the last number on the squared number – in this case 4 – and compare it to the last number of the answer.
70 * 70 will equal XXX0
71 * 71 will equal XXX1
72 * 72 will equal XXX4
73 * 73 will equal XXX9
74 * 74 will equal XXX(1)6
Therefore 72 is the answer. Check by multiplying it out.
The easiest way to narrow down a square root from a list is to look at the last number on the squared number – in this case 4 – and compare it to the last number of the answer.
70 * 70 will equal XXX0
71 * 71 will equal XXX1
72 * 72 will equal XXX4
73 * 73 will equal XXX9
74 * 74 will equal XXX(1)6
Therefore 72 is the answer. Check by multiplying it out.
Evaluate:
0.082
Evaluate:
0.082
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0.08 * 0.08
First square 8:
8 * 8 = 64
Then move the decimal four places to the left:
0.0064
0.08 * 0.08
First square 8:
8 * 8 = 64
Then move the decimal four places to the left:
0.0064
If $$\sqrt{x}$=3^2$ what is x?
If $$\sqrt{x}$=3^2$ what is x?
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Square both sides:
x = (32)2 = 92 = 81
Square both sides:
x = (32)2 = 92 = 81
Simplify.
Simplify.
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First step is to find perfect squares in all of our radicans.
After doing so you are left with 
*Just like fractions you can only add together coefficents with like terms under the radical. *
First step is to find perfect squares in all of our radicans.
After doing so you are left with
*Just like fractions you can only add together coefficents with like terms under the radical. *
Simplify:
Simplify:
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To combine radicals, they must have the same radicand. Therefore, we must find the perfect squares in each of our square roots and pull them out.
Now, we plug these equivalent expressions back into our equation and simplify:
To combine radicals, they must have the same radicand. Therefore, we must find the perfect squares in each of our square roots and pull them out.
Now, we plug these equivalent expressions back into our equation and simplify:
Simplify:
Simplify:
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Simplify each of the radicals by factoring out a perfect square:
Simplify each of the radicals by factoring out a perfect square:
Simplify the expression:
Simplify the expression:
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For each of the expressions, factor out a perfect square:
For each of the expressions, factor out a perfect square:
Add the square roots into one term:
Add the square roots into one term:
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In order to solve this problem we need to simplfy each of the radicals. By doing this we will get two terms that have the same number under the radical which will allow us to combine the terms.
In order to solve this problem we need to simplfy each of the radicals. By doing this we will get two terms that have the same number under the radical which will allow us to combine the terms.
Simplify:
Simplify:
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Remember that you treat square roots like you do variables in the sense that you just add the like factors. In this problem, the only set of like factors is the pair of 
values. Hence:
Do not try to simplify any further!
Remember that you treat square roots like you do variables in the sense that you just add the like factors. In this problem, the only set of like factors is the pair of
Do not try to simplify any further!
Simplify:
Simplify:
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Begin by simplifying your more complex roots:
This lets us rewrite our expression:
Do the basic multiplications of coefficients:
Reorder the terms:
Finally, combine like terms:
Begin by simplifying your more complex roots:
This lets us rewrite our expression:
Do the basic multiplications of coefficients:
Reorder the terms:
Finally, combine like terms:
(√27 + √12) / √3 is equal to
(√27 + √12) / √3 is equal to
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√27 is the same as 3√3, while √12 is the same as 2√3.
3√3 + 2√3 = 5√3
(5√3)/(√3) = 5
√27 is the same as 3√3, while √12 is the same as 2√3.
3√3 + 2√3 = 5√3
(5√3)/(√3) = 5
Divide and simplify. Assume all integers are positive real numbers.
Divide and simplify. Assume all integers are positive real numbers.
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There are two ways to solve this problem. First you can divide the numbers under the radical. Then simplify.
Example 1
Example 2
Find the square root of both numerator and denominator, simplifying as much as possible then dividing out like terms.
Both methods will give you the correct answer of 
.
There are two ways to solve this problem. First you can divide the numbers under the radical. Then simplify.
Example 1
Example 2
Find the square root of both numerator and denominator, simplifying as much as possible then dividing out like terms.
Both methods will give you the correct answer of
Simplify:
Simplify:
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To simplfy, we must first distribute the square root.
Next, we can simplify each of the square roots.
To simplfy, we must first distribute the square root.
Next, we can simplify each of the square roots.
Find the quotient:
Find the quotient:
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Simplify each radical:
Rationalize the denominator:
Simplify each radical:
Rationalize the denominator:
Find the quotient:
Find the quotient:
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Find the quotient:
There are two ways to approach this problem.
Option 1: Combine the radicals first, the reduce
Option 2: Simplify the radicals first, then reduce
Find the quotient:
There are two ways to approach this problem.
Option 1: Combine the radicals first, the reduce
Option 2: Simplify the radicals first, then reduce
How many integers from 20 to 80, inclusive, are NOT the square of another integer?
How many integers from 20 to 80, inclusive, are NOT the square of another integer?
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First list all the integers between 20 and 80 that are squares of another integer:
52 = 25
62 = 36
72 = 49
82 = 64
In total, there are 61 integers from 20 to 80, inclusive. 61 – 4 = 57
First list all the integers between 20 and 80 that are squares of another integer:
52 = 25
62 = 36
72 = 49
82 = 64
In total, there are 61 integers from 20 to 80, inclusive. 61 – 4 = 57
If x and y are integers and xy + y2 is even, which of the following statements must be true?
I. 3y is odd
II. y/2 is an integer
III. xy is even
If x and y are integers and xy + y2 is even, which of the following statements must be true?
I. 3y is odd
II. y/2 is an integer
III. xy is even
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In order for the original statement to be true, the 
and 
terms must be either both odd or both even. Looking at each of the statements individually,
I. States that 
is odd, but only odd values multiplied by 3 are odd. If 
was an even number, the result would be even. But 
can be either odd or even, depending on what 
equals. Thus this statement COULD be true but does not HAVE to be true.
II. States that 
is an integer, and since only even numbers are cleanly divided by 2 (odd values result in a fraction) this ensures that 
is even. However, 
can also be odd, so this is a statement that COULD be true but does not HAVE to be true.
III. For exponents, only the base value determines whether it is even or odd - it does not indicate the value of y at all. Only even numbers raised to any power are even, thus, this ensures that 
is even. But 
can be odd as well, so this statement COULD be true but does not HAVE to be true.
An example of two integers that will work violate conditions II and III is 
and 
.
, and even number.
is not an integer.
is not even.
Furthermore, any combination of 2 even integers will make the original statement true, and violate the Statement I.
In order for the original statement to be true, the
I. States that
II. States that
III. For exponents, only the base value determines whether it is even or odd - it does not indicate the value of y at all. Only even numbers raised to any power are even, thus, this ensures that
An example of two integers that will work violate conditions II and III is
Furthermore, any combination of 2 even integers will make the original statement true, and violate the Statement I.
Consider the inequality:
Which of the following could be a value of 
?
Consider the inequality:
Which of the following could be a value of
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Notice how $x^4$ is the greatest value. This often means that x is negative as $(-1)^n$=-1 when n is odd and $(-1)^n$=1 when n is even.
Let us examine the first choice, x=-$\frac{3}{4}$
$x^5$$=-$\frac{3^5$$}{4^5$}$=-$\frac{243}{1024}$> -$\frac{3}{4}$
This can only be true of a negative value that lies between zero and one.
Notice how $x^4$ is the greatest value. This often means that x is negative as $(-1)^n$=-1 when n is odd and $(-1)^n$=1 when n is even.
Let us examine the first choice, x=-$\frac{3}{4}$
$x^5$$=-$\frac{3^5$$}{4^5$}$=-$\frac{243}{1024}$> -$\frac{3}{4}$
This can only be true of a negative value that lies between zero and one.
If $5^{x}$$+5^{x}$$+5^{x}$$+5^{x}$$+5^{x}$$=5^{10}$, what is the value of x?
If $5^{x}$$+5^{x}$$+5^{x}$$+5^{x}$$+5^{x}$$=5^{10}$, what is the value of x?
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$5(5^{x}$$)=5^{10}$
$5^{1}$$(5^{x}$$)=5^{10}$
$5^{x+1}$$=5^{10}$
x=9
$5(5^{x}$$)=5^{10}$
$5^{1}$$(5^{x}$$)=5^{10}$
$5^{x+1}$$=5^{10}$
x=9
Simplify.
Simplify.
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Take the square root of both the top and bottom terms.
Simplify.
Take the square root of both the top and bottom terms.
Simplify.