GMAT Math : Arithmetic

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Example Question #43 : Understanding Powers And Roots

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 #1621 : Gmat Quantitative Reasoning

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

Possible Answers:

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

$\frac{1}{n}$

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

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

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 #45 : Understanding Powers And Roots

Simplify $\sqrt{5400}$

Possible Answers:

$40\sqrt{6}$

$30\sqrt{6}$

$10\sqrt{54}$

$30\sqrt{3}$

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 #41 : Understanding Powers And Roots

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:

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

(a) only

(a) and (b) only

(a) and (c) only

(b) and (c) 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:

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

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

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

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

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

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:

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

x^6z^5

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

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

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

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:

$\sqrt{yz}$

x^2yz^2

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

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

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:

$18\sqrt{41}$

$40\sqrt{164}$

$40\sqrt{41}$

$10\sqrt{656}$

$44\sqrt{41}$

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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Example Question #1 : Percents

330 is what percent of 11?

Possible Answers:

$\dpi{100} \small 3\%$

$\dpi{100} \small 3000\%$

$\dpi{100} \small 30\%$

$\dpi{100} \small 300\%$

$\dpi{100} \small 33\%$

Correct answer:

$\dpi{100} \small 3000\%$

Explanation:

This problem can be solved by the equation: $330 = 11x$ , where $\dpi{100} \small x$ is the answer in terms of a percentage. To solve for $\dpi{100} \small x$ both sides are divided by $\dpi{100} \small 11$ : $\frac{330 }{11}= x$ . This can be simplified to: $30 = x$ . Now $\dpi{100} \small 30$ is converted to a percentage to find the answer.

$\dpi{100} \small 30$ as a percentage is $\dpi{100} \small 3000\%$

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Example Question #1 : Percents

Three years ago, Anum invested $25,000 in a new mutual fund account. The value of the account increased by 15% during the first year, increased by 7% during the second year, and decreased by 15% during the third year. What is the approximate value of the account today?

Possible Answers:

$\dpi{100} \small \$ 31,000$

$\dpi{100} \small \$ 26,000$

$\dpi{100} \small \$ 25,000$

$\dpi{100} \small \$ 27,000$

$\dpi{100} \small \$ 35,000$

Correct answer:

$\dpi{100} \small \$ 26,000$

Explanation:

The first year increase of 15% can be represented as 1.15; the second year increase of 7% can be represented as 1.07; and the third year decrease of 15% can be represented as 0.85.

Multiply the original investment by each annual change.

25,000(1.15)(1.07)(0.85) = 26,148

approx. $26,000

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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #43 : Understanding Powers And Roots

Example Question #1621 : Gmat Quantitative Reasoning

Example Question #45 : Understanding Powers And Roots

Example Question #41 : Understanding Powers And Roots

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

Example Question #1 : Percents

Example Question #1 : Percents

All GMAT Math Resources

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