GMAT Math : Powers & Roots of Numbers

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Example Question #41 : Arithmetic

Evaluate $\left (625^{\frac{1}{2} } \right )^{\frac{1}{2} }$ .

Possible Answers:

$-\frac{1}{5}$

$\frac{1}{5}$

The expression is not a real number.

Correct answer:

Explanation:

$\left (625^{\frac{1}{2} } \right )^{\frac{1}{2} } =\left ( \sqrt{625}\right )^{\frac{1}{2} } = \left ( 25\right )^{\frac{1}{2} }= \sqrt{25} = 5$

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Example Question #22 : Understanding Powers And Roots

Evaluate $\left (-256^{\frac{1}{2} } \right )^{\frac{1}{2} }$ .

Possible Answers:

$-\frac{1}{4}$

The expression is not a real number.

$\frac{1}{4}$

Correct answer:

The expression is not a real number.

Explanation:

$256 ^{\frac{1}{2}} = \sqrt{256} = 16$ , so

$\left (-256^{\frac{1}{2} } \right )^{\frac{1}{2} } = \left (-16\right )^{\frac{1}{2} } = \sqrt{-16}$

However, has no real square root. The expression is not equal to a real number.

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Example Question #23 : Powers & Roots Of Numbers

Which of the following is equivalent to $\sqrt[3]{\sqrt[4]{x}}$ ?

You may assume to be positive.

Possible Answers:

$-\sqrt{x}$

$\sqrt[12]{x}$

$\sqrt{x}$

$\sqrt[7]{x}$

$\sqrt[64]{x}$

Correct answer:

$\sqrt[12]{x}$

Explanation:

Convert the roots to fractional exponents, then back, as follows:

$\sqrt[3]{\sqrt[4]{x}} = \sqrt[3]{ x^{\frac{1}{4}}} = \left ( x^{\frac{1}{4}} \right ) ^{\frac{1}{3}}= x ^{\frac{1}{4} \cdot \frac{1}{3}} = x^{\frac{1}{12}}= \sqrt[12]{x}$

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Example Question #24 : Powers & Roots Of Numbers

Evaluate $\left (1,296^{\frac{1}{3} } \right )^{\frac{1}{2} }$ .

Possible Answers:

The correct answer is not among the other responses.

1,296

$216 \sqrt[3]{6}$

$\sqrt[5]{1,296}$

$\sqrt[3]{36}$

Correct answer:

$\sqrt[3]{36}$

Explanation:

$1,296 = 6 ^{4}$ , so

$\left (1,296^{\frac{1}{3} } \right )^{\frac{1}{2} } = \left [ \left (6 ^{4 } \right )^{\frac{1}{3} } \right ]^{\frac{1}{2} } = 6 ^{4 \cdot \frac{1}{3} \cdot \frac{1}{2}} =6 ^{ \frac{2}{3} }$

For all integers M,N and all positive bases ,

$a^{ \frac{M}{N}} = \sqrt[N]{a^{M}}$ by definition. So set

a = 6, M = 2, N = 3 :

$\left (1,296^{\frac{1}{3} } \right )^{\frac{1}{2} }=6 ^{ \frac{2}{3} } = \sqrt[3]{6 ^{2}} = \sqrt[3]{36}$

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Example Question #25 : Powers & Roots Of Numbers

Simplify by rationalizing the denominator:

$\frac{1+\sqrt{2}}{1+\sqrt{3}}$

Possible Answers:

$\frac{-1+\sqrt{2}- \sqrt{3}+\sqrt{6} } { 2}$

$\frac{-1 +\sqrt{6} } { 2}$

$\frac{-1+\sqrt{2}- \sqrt{3}+\sqrt{6} } { 4}$

$\frac{-1- \sqrt{2} + \sqrt{3}+\sqrt{6} } { 2}$

$\frac{-1 +\sqrt{6} } { 4}$

Correct answer:

$\frac{-1- \sqrt{2} + \sqrt{3}+\sqrt{6} } { 2}$

Explanation:

Multiply both numerator and denominator by the conjugate of the denominator, which is $1-\sqrt{3}$ :

$\frac{1+\sqrt{2}}{1+\sqrt{3}}$

$=\frac{(1+\sqrt{2})(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}$

$=\frac{ 1 \cdot 1-1 \cdot \sqrt{3}+ \sqrt{2} \cdot 1-\sqrt{2} \cdot \sqrt{3}} { 1^{2}-(\sqrt{3})^{2}}$

$=\frac{1- \sqrt{3}+ \sqrt{2} -\sqrt{6} } { 1-3}$

$=\frac{1+ \sqrt{2} - \sqrt{3}-\sqrt{6} } { -2}$

$=\frac{-1- \sqrt{2} + \sqrt{3}+\sqrt{6} } { 2}$

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Example Question #26 : Powers & Roots Of Numbers

If $\sqrt[3]{4^{6}.8^{3}} = 2^{m}$ , what is the value of ?

Possible Answers:

Correct answer:

Explanation:

We need to put the expression under the format of 2 to the power of m:

$\sqrt[3]{4^{6}.8^{3}} = \sqrt[3]{(2^{2})^{6}.(2^{3})^{3}} = \sqrt[3]{2^{12}.2^{9}}$

$\sqrt[3]{2^{21}}=(2^{21})^{\frac{1}{3}}=2^{\frac{21}{3}}=2^7$

Therefore the right answer is 7.

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Example Question #27 : Powers & Roots Of Numbers

$(\sqrt{75}+\sqrt{108})^{\sqrt[3]{8}}=$

Possible Answers:

363

$183+22\sqrt{3}$

$194\sqrt{3}$

$183+60\sqrt{3}$

183

Correct answer:

363

Explanation:

$(\sqrt{75}+\sqrt{108})^{\sqrt[3]{8}}=(\sqrt{75}+\sqrt{108})^{2}$

$=(\sqrt{75})^{2}+(\sqrt{108})^{2}+2\times \sqrt{75}\times \sqrt{108}$

$=75+108+2\times \sqrt{3(5^{2})}\times \sqrt{3(6^{^{2}})}$

$=75+108+2\times5\sqrt{3}\times6\sqrt{3}$

$=183+60(\sqrt{3})^{2}$

$=183+60\times3$

=183+180

=363

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Example Question #28 : Powers & Roots Of Numbers

Evaluate $24 ^{\frac{2}{3}}$ , expressing the result as a simplified radical if applicable.

Possible Answers:

$4 \sqrt[3]{9}$

$48\sqrt{6}$

The correct answer is not among the other responses.

Correct answer:

$4 \sqrt[3]{9}$

Explanation:

For all integers M,N and all positive bases ,

$a^{ \frac{M}{N}} = \sqrt[N]{a^{M}}$ by definition.

Set a = 24, M = 2, N = 3 :

$24^{ \frac{2}{3}} = \sqrt[3]{24^{2}} = \sqrt[3]{576} = \sqrt[3]{64} \cdot \sqrt[3]{9}= 4 \sqrt[3]{9}$

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Example Question #29 : Powers & Roots Of Numbers

Which of the following numbers is irrational?

Possible Answers:

$\sqrt{16}$

$\sqrt{4}$

$\sqrt[3]{8}$

$\sqrt{2}$

None of the other answers.

Correct answer:

$\sqrt{2}$

Explanation:

A square root of a number is irrational if the number inside the square root is not a perfect square. Since 2 is not a perfect square, $\sqrt{2}$ is irrational

$\sqrt[3]{8}$ is rational, because is a perfect cube. ( $2\times2\times2 = 8$ )

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Example Question #1 : How To Factor A Common Factor Out Of Squares

Which of the following expression is equal to

$\sqrt{(45)(9)+(27)(36)}$

Possible Answers:

$7\sqrt{15}$

$9\sqrt{17}$

$9\sqrt{15}$

$9\sqrt{19}$

$7\sqrt{17}$

Correct answer:

$9\sqrt{17}$

Explanation:

$\sqrt{(45)(9)+(27)(36)}$

When simplifying a square root, consider the factors of each of its component parts:

$\sqrt{(3^2)\times5\times(3^{2})+(3^{3})(2^2)(3^2)}$

Combine like terms:

$\sqrt{5(3^4)+(2^2)(3^5)}$

Remove the common factor, 3^4 :

$\sqrt{(3^4)\times(5+3\times(2^2))}$

Pull the $\sqrt{3^4}$ outside of the equation as 3^2 :

$(3^2)\sqrt{5+12}=9\sqrt{17}$

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GMAT Math : Powers & Roots of Numbers

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #41 : Arithmetic

Example Question #22 : Understanding Powers And Roots

Example Question #23 : Powers & Roots Of Numbers

Example Question #24 : Powers & Roots Of Numbers

Example Question #25 : Powers & Roots Of Numbers

Example Question #26 : Powers & Roots Of Numbers

Example Question #27 : Powers & Roots Of Numbers

Example Question #28 : Powers & Roots Of Numbers

Example Question #29 : Powers & Roots Of Numbers

Example Question #1 : How To Factor A Common Factor Out Of Squares

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