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SAT Math : How to simplify square roots

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Example Questions

Example Question #51 : Simplifying Square Roots

Simplify:

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

Possible Answers:

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

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

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

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

$-\frac{7}{5}$

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 #83 : Basic Squaring / Square Roots

Simplify: $\sqrt{450}$

Possible Answers:

$5\sqrt{18}$

$3\sqrt{50}$

$15\sqrt2$

$10\sqrt4$

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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SAT Math : How to simplify square roots

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #51 : Simplifying Square Roots

Example Question #82 : Basic Squaring / Square Roots

Example Question #83 : Basic Squaring / Square Roots

All SAT Math Resources

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