GMAT Math : Powers & Roots of Numbers

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Example Question #1 : Powers & Roots Of Numbers

Solve: $\frac{(0.5)^{6}}{(0.5)^{9}}$

Possible Answers:

$\dpi{100} \small 0.125$

$\dpi{100} \small 125$

$\dpi{100} \small 0.25$

$\dpi{100} \small 8$

$\dpi{100} \small 4$

Correct answer:

$\dpi{100} \small 8$

Explanation:

Solve $\dpi{100} \small : \frac{0.5^{6}}{0.5^{9}} = 0.5^{6-9}= 0.5^{-3}=\frac{1}{0.5^{3}}=\frac{1}{0.125} = 8$

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Example Question #1 : Understanding Powers And Roots

$\frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} =$

Possible Answers:

$\dpi{100} \small cannot\ be\ simplified$

$\frac{yz}{x}$

$\frac{1}{x^{2}}$

$\frac{y^{2}z}{x}$

$\frac{yz}{x^{2}}$

Correct answer:

$\frac{y^{2}z}{x}$

Explanation:

$\frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} = \frac{y^{3-1}z^{4-3}}{x^{3-2}} = \frac{y^{2}z}{x}$

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Example Question #3 : Powers & Roots Of Numbers

Solve: $\small (\sqrt{5}+\sqrt{4})^2$

Possible Answers:

$\small 20$

$\small 9+4\sqrt5$

$\small 20+4\sqrt5$

$\small 9$

Correct answer:

$\small 9+4\sqrt5$

Explanation:

First, FOIL:

$\small (\sqrt5+\sqrt4)^2=(\sqrt5+\sqrt4)(\sqrt5+\sqrt4)=5+\sqrt20+\sqrt20+4$

$\small =9+2\sqrt20$

Factor out $\small \sqrt4$

$\small =9+4\sqrt5$

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Example Question #1 : Understanding Powers And Roots

Solve: $\small \frac{10^9-10^7}{99}$

Possible Answers:

$\small \frac{10^7}{99}$

$\small \frac{10}{99}$

$\small \frac{100}{99}$

$\small 10^7$

Correct answer:

$\small 10^7$

Explanation:

First factor. $\small \frac{10^9-10^7}{99}\ =\ \frac{(10^7)(10^2-1)}{99}$

Simplify. $\small \frac{(10^7)(10^2-1)}{99}\ =\ \frac{(10^7)(100-1)}{99}\ =\ \frac{(10^7)(99)}{99}\ =\ 10^7$

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Example Question #1 : Understanding Powers And Roots

If $\small x^2=8$ , what is $\small x^4?$

Possible Answers:

124

Correct answer:

Explanation:

$\small x^4=(x^2)^2=(8)^2=64$

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Example Question #6 : Powers & Roots Of Numbers

Evaluate: $11^{100} - 11^{99}$

Possible Answers:

$10 \cdot 11^{99}$

$11^{99}$

$11^{98}$

$10 \cdot 11^{98}$

Correct answer:

$10 \cdot 11^{99}$

Explanation:

$11^{100} - 11^{99} = 11 \cdot 11^{99} - 1\cdot 11^{99} = (11-1) \cdot 11^{99} = 10 \cdot 11^{99}$

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Example Question #7 : Powers & Roots Of Numbers

Simplify this expression as much as possible:

$\sqrt{9x}+\sqrt{12x}+\sqrt{27x}$

Possible Answers:

$8\sqrt{x}$

$8\sqrt{3x}$

$\sqrt{3x}+\sqrt{15x}$

$3\sqrt{x}+5\sqrt{3x}$

The expression cannot be simplified further

Correct answer:

$3\sqrt{x}+5\sqrt{3x}$

Explanation:

$\sqrt{9x}+\sqrt{12x}+\sqrt{27x}$

$=\sqrt{9}\cdot \sqrt{x}+\sqrt{4}\cdot \sqrt{3x}+\sqrt{9}\cdot \sqrt{3x}$

$=3 \sqrt{x}+2 \sqrt{3x}+3 \sqrt{3x}$

$=3 \sqrt{x}+\left (2 +3 \right )\sqrt{3x}$

$=3 \sqrt{x}+5\sqrt{3x}$

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Example Question #1 : Understanding Powers And Roots

If the side length of a cube is tripled, how does the volume of the cube change?

Possible Answers:

Volume becomes 3 times larger.

Not enough informatin is given.

Volume becomes 27 times larger.

Volume becomes 9 times larger.

The volume doesn't change.

Correct answer:

Volume becomes 27 times larger.

Explanation:

The equation for the volume of a cube is $L^{3}$ . If the length is tripled, it becomes $(3L)^{3}$ , and $3^{3}=27$ , so the volume increases by 27 times the original size.

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Example Question #9 : Powers & Roots Of Numbers

Simplify $\sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$

Possible Answers:

Correct answer:

Explanation:

This can either be done by brute force (slow) or by recognizing the properties of roots and exponents (fast). Roots are simply fractional exponents: $\sqrt{x}=x^{\frac{1}{2}}$ , $\sqrt[3]{x}=x^{\frac{1}{3}}$ , etc. so they can be done in any order.

So we see a cube root, we can immediately cancel that with the exponent of 3. taking us from here: $\sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$ to $\sqrt{\sqrt[4]{(4^2)^2}}$ . We now simplify (4^2)^2 = 4^4 to get $\sqrt{\sqrt[4]{(4^4)}} = \sqrt4 = 2$

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Example Question #10 : Powers & Roots Of Numbers

In the sequence 1, 3, 9, 27, 81, … , each term after the first is three times the previous term. What is the sum of the 9th and 10th terms in the sequence?

Possible Answers:

$6(3)^{9}$

$6(3)^{8}$

$(3)^{10}$

$4(3)^{8}$

$4(3)^{9}$

Correct answer:

$4(3)^{8}$

Explanation:

We can rewrite the sequence as $(3)^{0}$ , $(3)^{1}$ , $(3)^{2}$ , $(3)^{3}$ , $(3)^{4}$ , … ,

and we can see that the 9th term in the sequence is $(3)^{8}$ and the 10th term in the sequence is $(3)^{9}$ . Therefore, the sum of the 9th and 10th terms would be

$3^{8}+3^{9}=3^{8}\cdot \left ( 1+3 \right )=4\cdot 3^{8}$

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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 #1 : Powers & Roots Of Numbers

Example Question #1 : Understanding Powers And Roots

Example Question #3 : Powers & Roots Of Numbers

Example Question #1 : Understanding Powers And Roots

Example Question #1 : Understanding Powers And Roots

Example Question #6 : Powers & Roots Of Numbers

Example Question #7 : Powers & Roots Of Numbers

Example Question #1 : Understanding Powers And Roots

Example Question #9 : Powers & Roots Of Numbers

Example Question #10 : Powers & Roots Of Numbers

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