GMAT Math : Powers & Roots of Numbers

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Example Questions

Example Question #41 : Understanding Powers And Roots

How can $4^{n}\cdot\frac{1}{2^{n}}\cdot5^{n}$ be rewritten?

Possible Answers:

$4^{n-1}\cdot 10$

$4^{2n-2}\cdot 5^{n}$

$10^{n}$

$4^{n-2}\cdot 5^{n}$

$4^{n-1}\cdot 5^{n}$

Correct answer:

$10^{n}$

Explanation:

$4^{n}\cdot\frac{1}{2^{n}}\cdot5^{n}$

Multiply the terms:

$\frac{4^n\cdot5^n}{2^n}$

$\frac{20^n}{2^n}$

10^n

Alternatively, this problem can be solved by rewriting the fraction $\frac{1}{2^n}$ as a negative exponent:

$4^n\cdot2^{-n}\cdot5^n$

Then, we can rewrite 4^n as (2^2)^n . Using the rules of exponents, we multiply these exponents to get $2^{2n}$ :

$2^{2n}\cdot2^{-n}\cdot5^n$

At this point, we can combine the two terms that have a base of :

$2^{2n-n}\cdot5^n$

$2^n\cdot5^n$

$10^{n}$

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Example Question #1621 : Gmat Quantitative Reasoning

Which of the following expressions is equal to $5^{n}-5^{n+1}$ ?

Possible Answers:

$\frac{1}{5}$

$-5^{n}$

$5^{n}\cdot-4$

$5^{\frac{n}{n+1}}$

Correct answer:

$5^{n}\cdot-4$

Explanation:

As soon as we see that this is a difference of exponential factors that have a common base of 5, we should ask ourselves whether we can factor any common factors out of the terms. We can rewrite the expression using the rules of exponents to reveal a common factor of $5^{n}$ :

$5^{n}-5^{n+1}$

$5^n-(5^n\cdot5^1)$

At this point, we can factor out 5^n :

$5^n(1-(1\cdot5^1))$

5^n(1-5)

$5^n\cdot-4$

$5^n\cdot-4$ is one of the listed answers, so we have arrived at the correct answer.

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Example Question #71 : Arithmetic

Which of the following answers is closest to $\frac{10^{8}-10^{3}}{10^{7}-10^{4}}$ ?

Possible Answers:

1000

100

Correct answer:

Explanation:

Here, we should factor out the smaller powers of to be able to simplify: $\frac{10^{8}-10^{3}}{10^{7}-10^{4}} = \frac{10^{3}(10^{5}-1)}{10^{4}(10^{3}-1)}$

Since these powers of are all quite large, we can approximate the answer by dropping the two (-1) and we get $\frac{10^{8}}{10^{7}}$ . When dividing terms with exponents, the exponents should be subtracted, so our expression can be rewritten as 10^1 , which is equal to .

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Example Question #71 : Arithmetic

How can $\frac{\sqrt{n}}{n}$ be rewritten?

Possible Answers:

$\frac{n}{n^{2}}$

$\frac{n}{\sqrt{n}}$

$\frac{1}{n}$

$n^{-\frac{1}{2}}$

Correct answer:

$n^{-\frac{1}{2}}$

Explanation:

By simply writing $\sqrt n$ as a power, we get $n^{\frac{1}{2}}$ , making the expression $\frac{n^\frac{1}{2}}{n}$ , or $\frac{n^\frac{1}{2}}{n^1}$ .

When dividing terms with exponents, the exponents should be subtracted, so this can be written as $n^{\frac{1}{2}-1}$ or $n^{-\frac{1}{2}}$ .

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Example Question #73 : Arithmetic

Simplify $\sqrt{5400}$

Possible Answers:

$30\sqrt{3}$

$30\sqrt{6}$

$40\sqrt{6}$

$10\sqrt{54}$

Correct answer:

$30\sqrt{6}$

Explanation:

Our first step will be to factor the expression:

$\sqrt{5400} = \sqrt{54 \cdot 100} = 10\sqrt{54}$

54 and 100 are our first factors. We can then take the square root of 100 and move it to the outside of the radical. We can then factor 54 and simplify the square root of 9.

$10\sqrt{54 }= 10\sqrt{9 \cdot 6} = 10 \cdot3\sqrt{6} = 30\sqrt{6}$

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Example Question #74 : Arithmetic

and are positive integers, with M > N . Which of the following must also be an integer?

(a) $\sqrt{M^{2}-2MN+N^{2}}$

(b) $\sqrt{M^{2}+4MN+4N^{2}}$

Possible Answers:

(b) and (c) only

(a) and (c) only

(a) and (b) only

(a), (b), and (c)

(a) only

Correct answer:

(a), (b), and (c)

Explanation:

All three radicands can be seen to be perfect square trinomials:

$M^{2}-2MN+N^{2}= (M-N)^{2}$

$M^{2}+4MN+4N^{2} = M^{2}+2 \cdot M \cdot 2N+ \left (2N \right )^{2} = (M + 2N)^{2}$

$9M^{2}+6MN+N^{2} = \left ( 3M\right )^{2}+ 2 \cdot 3M \cdot N + N^{2} = (3M+N)^{2}$

Therefore (since we are given that M > N > 0 , we do not need absolute value symbols):

(a) $\sqrt{M^{2}-2MN+N^{2}} = \sqrt{(M - N)^{2}} = M - N$

(b) $\sqrt{M^{2}+4MN+4N^{2}} = \sqrt{(M + 2N)^{2}} = M + 2N$

Since the integers are closed under addition and subtraction, all three expressions are integers.

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Example Question #47 : Understanding Powers And Roots

and are positive integers, with M > N . Which of the following must also be an integer?

Possible Answers:

$\sqrt{M^{2}+2MN+N^{2}}$

$\sqrt{M^{2}-N^{2}}$

None of the expressions given in the other choices must be integers.

$\sqrt{M^{2}+ MN+N^{2}}$

$\sqrt{M^{2}+ N^{2}}$

Correct answer:

$\sqrt{M^{2}+2MN+N^{2}}$

Explanation:

For three of the choices, we can produce examples for each that show that they can be nonintegers. For example, if M = 2 and N = 1 :

$\sqrt{M^{2}+ MN+N^{2}} = \sqrt{2^{2}+ 2 \cdot 1 +1^{2}} = \sqrt{4+ 2 +1 }= \sqrt{7}$

$\sqrt{M^{2}+ N^{2}} = \sqrt{2^{2}+ 1^{2}} = \sqrt{4+ 1 } = \sqrt{5}$

$\sqrt{M^{2}- N^{2}} = \sqrt{2^{2}- 1^{2}} = \sqrt{4- 1 } = \sqrt{3}$

3, 5, and 7 are not perfect square integers, so none of the three expressions are integers.

The radicand of $\sqrt{M^{2}+2MN+N^{2}}$ , on the other hand, can be recognized as a perfect square trinomial, which is factorable as

$M^{2}+2MN+N^{2} = (M + N)^{2}$

$\sqrt{M^{2}+2MN+N^{2}} = \sqrt{ (M + N)^{2}} = M + N$ (we do not need absolute value bars since both and are positive).

As the sum of integers, the expression must itself be an integer.

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Example Question #41 : Powers & Roots Of Numbers

Evalute:

$(x^4y^{-2}z^3})^2$

Possible Answers:

$x^8y^{-4}z^6$

x^6z^5

$x^{8}y^{4}z$

$x^2y^{-4}z$

$x^{8}y^{-6}z^{10}$

Correct answer:

$x^8y^{-4}z^6$

Explanation:

To simplify exponents raised to another exponent, simply distribute the exponent and multiply the values.

$(x^4y^{-2}z^3})^2=x^{4*2}y^{-2*2}z^{3*2}=x^8y^{-4}z^6$

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Example Question #42 : Powers & Roots Of Numbers

Simplify completely:

$\sqrt{x^4y^3z^5}$

Possible Answers:

x^2yz^2

$\sqrt{yz}$

$xyz\sqrt{x^2yz^3}$

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

Correct answer:

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

Explanation:

To solve, simply pull out as many squared factos as you can, leaving the rest behind. Thus, the answer is:

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

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Example Question #43 : Powers & Roots Of Numbers

Simplify $\sqrt{65,600}$

Possible Answers:

$44\sqrt{41}$

$18\sqrt{41}$

$10\sqrt{656}$

$40\sqrt{164}$

$40\sqrt{41}$

Correct answer:

$40\sqrt{41}$

Explanation:

To simplify this root, we must start by pulling out as many perfect squares as possible:

$\sqrt{65,600} = \sqrt{656}\cdot\sqrt{100} = 10\sqrt{656}$

$10\sqrt{656} = 10\cdot \sqrt{16} \cdot \sqrt{41} = 10\cdot4\cdot \sqrt{41} = 40\sqrt{41}$

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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 : Understanding Powers And Roots

Example Question #1621 : Gmat Quantitative Reasoning

Example Question #71 : Arithmetic

Example Question #71 : Arithmetic

Example Question #73 : Arithmetic

Example Question #74 : Arithmetic

Example Question #47 : Understanding Powers And Roots

Example Question #41 : Powers & Roots Of Numbers

Example Question #42 : Powers & Roots Of Numbers

Example Question #43 : Powers & Roots Of Numbers

All GMAT Math Resources

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