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SAT Math : Simplifying Square Roots

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Example Questions

Example Question #82 : Basic Squaring / Square Roots

Simplify:

$\frac{7}{\sqrt{5}}$

Possible Answers:

$\frac{7\sqrt{5}-21}{4}$

$\frac{3-7\sqrt{5}}{4}$

$-\frac{7}{5}$

$\frac{5-7\sqrt{5}}{2}$

$\frac{7\sqrt{5}}{5}$

Correct answer:

$\frac{7\sqrt{5}}{5}$

Explanation:

To eliminate a radical expression, we need to multiply top and bottom by the denominator.

Then simplify if necessary.

$\\\frac{7}{\sqrt{5}}\cdot\frac{\sqrt{5}}{\sqrt{5}} \\ \\ =\frac{7\sqrt{5}}{5}$

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Example Question #82 : Basic Squaring / Square Roots

If x>0 , simplify $\sqrt{16x^4-48x^2}$

Possible Answers:

$4x^2\sqrt{-2}$

$4x\sqrt{x^2-3}$

4x^2-7x

$4x^2\sqrt{x^2-3}$

$4\sqrt{x^4-3x^2}$

Correct answer:

$4x\sqrt{x^2-3}$

Explanation:

In order to simplify the expression, we must see if we can factor out any perfect squares.

In this case, we can factor out 16x².

$\sqrt{16x^2(x^2-3)}$

Because of the rules of square roots, this can be rewritten as:

$\sqrt{16x^2}\sqrt{x^2-3}$

Because x>0 , $\sqrt{16x^2}$ simplifies to .

Thus, we can rewrite as $4x\sqrt{x^2-3}$

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Example Question #51 : How To Simplify Square Roots

Simplify: $\sqrt{450}$

Possible Answers:

$5\sqrt{18}$

$15\sqrt2$

$10\sqrt4$

$3\sqrt{50}$

Correct answer:

$15\sqrt2$

Explanation:

To simplify square roots, factor the inside completely to find perfect squares:

$450=45\cdot10=9\cdot5\cdot5\cdot2=3^2\cdot5^2\cdot2$

Then separate into several roots being multiplied:

$\sqrt{450}=\sqrt{3^2\cdot5^2\cdot2}=\sqrt{3^2}\cdot\sqrt{5^2}\cdot\sqrt2$

Simplify:

$\sqrt{3^2}\cdot\sqrt{5^2}\cdot\sqrt2=3 \cdot 5 \cdot \sqrt2 = 15 \sqrt2$

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Example Question #1 : How To Find The Common Factors Of Squares

If m and n are postive integers and 4^m = 2ⁿ, what is the value of m/n?

Possible Answers:

1/2

Correct answer:

1/2

Explanation:

2²= 4. Also, following the rules of exponents, 4¹= 1.
One can therefore say that m = 1 and n = 2.
The question asks to solve for m/n. Since m = 1 and n = 2, m/n = 1/2.

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Example Question #11 : Simplifying Square Roots

Simplify:

$\sqrt[2]{24,300}$

Possible Answers:

$9\sqrt[2]{300}$

$90\sqrt[2]{3}$

$90\sqrt[2]{270}$

$900\sqrt[2]{3}$

$10\sqrt[2]{243}$

Correct answer:

$90\sqrt[2]{3}$

Explanation:

$\sqrt[2]{24,300}$

When simplifying the square root of a number that may not have a whole number root, it's helpful to approach the problem by finding common factors of the number inside the radicand. In this case, the number is 24,300.

What are the factors of 24,300?

24,300 can be factored into:

$24,300: 243\cdot 100$

When there are factors that appear twice, they may be pulled out of the radicand. For instance, 100 is a multiple of 24,300. When 100 is further factored, it is $10^{2}$ (or 10x10). However, 100 wouldn't be pulled out of the radicand, but the square root of 100 because the square root of 24,300 is being taken. The 100 is part of the24,300. This means that the problem would be rewritten as: $10\sqrt[2]{243}$

But 243 can also be factored: $24,300: (3\cdot 81)\cdot100$
$24,300: {\color{Orange} 9}\cdot{\color{Orange} 9}\cdot {\color{Cyan} 10}\cdot {\color{cyan} 10}\cdot 3$
Following the same principle as for the 100, the problem would become
$9\cdot 10\sqrt[2]{3}$ because there is only one factor of 3 left in the radicand. If there were another, the radicand would be lost and it would be 9*10*3.
9 and 10 may be multiplied together, yielding the final simplified answer of
$90\sqrt[2]{3}$

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Example Question #1 : How To Find The Common Factors Of Squares

$\sqrt{180} + \sqrt{125} = ?$

Possible Answers:

17.5

$11\sqrt{10}$

$11\sqrt{5}$

$\sqrt{305}$

25.0

Correct answer:

$11\sqrt{5}$

Explanation:

To solve the equation $\sqrt{180} + \sqrt{125} = ?$ , we can first factor the numbers under the square roots.

$\sqrt{2\cdot 2\cdot 3\cdot 3\cdot 5} + \sqrt{5\cdot 5\cdot 5} = ?$

When a factor appears twice, we can take it out of the square root.

$2\cdot 3\sqrt{5} + 5\sqrt{5} = ?$

$6\sqrt{5} + 5\sqrt{5} = ?$

Now the numbers can be added directly because the expressions under the square roots match.

$6\sqrt{5} + 5\sqrt{5} = 11\sqrt{5}$

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SAT Math : Simplifying Square Roots

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #82 : Basic Squaring / Square Roots

Example Question #82 : Basic Squaring / Square Roots

Example Question #51 : How To Simplify Square Roots

Example Question #1 : How To Find The Common Factors Of Squares

Example Question #11 : Simplifying Square Roots

Example Question #1 : How To Find The Common Factors Of Squares

All SAT Math Resources

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