GMAT Math : Powers & Roots of Numbers

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Example Questions

Example Question #1 : Dsq: Understanding Powers And Roots

is a real number. Is positive, negative, or zero?

Statement 1: 2x - 5 > 0

Statement 2: $\frac{1}{2}x^{3} -1> 0$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If 2x - 5 > 0 , then 2x > 5 , and $x > \frac{5}{2} > 0$ , so must be positive.

If $\frac{1}{2}x^{3} -1> 0$ , then $\frac{1}{2}x^{3} > 1$ , $x^{3} > 2$ . and $x> \sqrt[3]{2}$ , so again, must be positive. Either statement is enough to answer the question in the affirmative.

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Example Question #1 : Powers & Roots Of Numbers

Simplify this expression as much as possible:

$\sqrt[3]{16} + \sqrt[3]{128} - \sqrt[3]{54}$

Possible Answers:

$4 + 4\sqrt[3]{2} - 3 \sqrt[3]{6}$

$3 \sqrt[3]{2}$

$9 \sqrt[3]{2}$

$\sqrt[3]{198}$

The expression is already simplified

Correct answer:

$3 \sqrt[3]{2}$

Explanation:

$\sqrt[3]{16} + \sqrt[3]{128} - \sqrt[3]{54}$

$= \sqrt[3]{8} \cdot \sqrt[3]{2} + \sqrt[3]{64} \cdot \sqrt[3]{2} - \sqrt[3]{27} \cdot \sqrt[3]{2}$

$= 2 \sqrt[3]{2} +4 \sqrt[3]{2} -3 \sqrt[3]{2}$

$=\left ( 2+4-3 \right ) \sqrt[3]{2} = 3 \sqrt[3]{2}$

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Example Question #2 : Dsq: Understanding Powers And Roots

Imagine an integer such that the units digit of is greater than 5. What is the units digit of ?

(1) The units digit of is the same as the units digit of $y^{2}$ .

(2) The units digit of is the same as the units digit of $y^{3}$ .

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.

BOTH statements TOGETHER are not sufficient to answer the question.

EACH statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Explanation:

(1) The only single-digit integer greater than 5 whose unit digit of its square term is equal to itself is 6. This statement is sufficient.

(2) There are two single-digit integers where the unit digit of the cubed term is equal to the integer itself: 6 and 9. This statement is insufficient.

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Example Question #301 : Arithmetic

What is the value of twelve raised to the fourth power?

Possible Answers:

1,728

144

28,380

20,736

Correct answer:

20,736

Explanation:

"Twelve raised to the fourth power" is 12⁴. If you can translate the words into their mathematical counterpart, you're done, because the actual calculation should be done by your calculator. It will tell you that $12\cdot 12\cdot 12\cdot 12=20,736$ . There is not enough time on the test for you to try to do this by hand.

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Example Question #4 : Dsq: Understanding Powers And Roots

Calculate the fifth root of :

(1) The square root of is 3125 .

(2) The tenth root of is .

Possible Answers:

Both statements TOGETHER are not sufficient.

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient.

Each statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Correct answer:

Each statement ALONE is sufficient.

Explanation:

Using Statement (1):

$m^{\frac{1}{5}}= (m^{\frac{1}{2}})^{\frac{2}{5}}=(3125)^{\frac{2}{5}}=25$

Statement (1) ALONE is SUFFICIENT.

Using Statement (2):

$m^{\frac{1}{5}}= (m^{\frac{1}{10}})^{2}=(5)^{2}=25$

Statement (2) ALONE is SUFFICIENT.

Therefore EACH Statement ALONE is sufficient.

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Example Question #5 : Dsq: Understanding Powers And Roots

is a positive real number. True or false: is a rational number.

Statement 1: $N ^{3}$ is an irrational number.

Statement 2: $N ^{4}$ is an irrational number.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

An integer power of a rational number, being a product of rational numbers, must itself be rational. Either statement alone asserts that such a power is irrational, so conversely, either statement alone proves irrational.

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Example Question #6 : Powers & Roots Of Numbers

N > 1 . True or false: is rational.

Statement 1: $\left (N + 1 \right )^{2}$ is rational.

Statement 2: $\left (N - 1 \right )^{2}$ is rational.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone is not enough to prove is or is not rational. Examples:

If N = 2 , then $\left (N+ 1 \right )^{2} = \left (2+ 1 \right )^{2} = 3^{2} = 9$

If $N = \sqrt{2} - 1$ , then $\left (N+ 1 \right )^{2} = \left ( \sqrt{2}-1 + 1 \right )^{2} =( \sqrt{2})^{2} = 2$

In both cases, $\left (N + 1 \right )^{2}$ is rational, but in one case, is rational and in the other, is irrational.

A similar argument demonstrates Statement 2 to be insufficient.

Assume both statements are true. $\left (N + 1 \right )^{2}$ and $\left (N - 1 \right )^{2}$ are rational, so their difference is as well:

$\left (N + 1 \right )^{2} - \left (N - 1 \right )^{2}$

$=\left ( N ^{2}+ 2N + 1 \right ) - \left ( N ^{2}- 2N + 1 \right )$

$= N ^{2}- N ^{2}+ 2N +2N + 1 -1$

= 4N

is rational, so by closure under division, $4N \div 4 = N$ is rational.

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Example Question #6 : Dsq: Understanding Powers And Roots

N > 1 . True or false: is rational.

Statement 1: $\sqrt{N+1}$ is irrational.

Statement 2: $\sqrt{N-1}$ is rational.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 alone is not enough to prove rational or irrational. Examples:

If N = 4 , then $\sqrt{N+1} = \sqrt{4+1} = \sqrt{5}$

If $N = \pi -1$ , then $\sqrt{N+1} = \sqrt{(\pi -1)+1} = \sqrt{\pi}$

In both cases, $\sqrt{N+1}$ is irrational, but in only one case, is rational.

Assume Statement 2 alone. $\sqrt{N-1}$ is rational, so, by closure of the rational numbers under multiplication,

$\sqrt{N-1} \cdot \sqrt{N-1} = N - 1$ is rational. The rationals are closed under addition, so the sum

N - 1+ 1 = N

is rational.

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Example Question #2 : Dsq: Understanding Powers And Roots

is a positive real number. True or false: $\sqrt{N}$ is a rational number.

Statement 1: is irrational.

Statement 2: $\sqrt[4]{N}$ is irrational.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

If $\sqrt{N}$ is rational, then, since the product of two rational numbers is rational, $\sqrt{N} \cdot \sqrt{N} = N$ is rational. If Statement 1 alone is assumed, then, since is irrational, $\sqrt{N}$ must be irrational.

Assume Statement 2 alone, and note that

$\left (\sqrt[4]{N} \right )^{2}= \left (N ^{\frac{1}{4}} \right )^{2}= N ^{\frac{1}{4} \cdot 2} = N ^{\frac{1}{2} } = \sqrt{N}$

In other words, $\sqrt[4]{N}$ is the square root of $\sqrt{N}$ . Since both rational and irrational numbers have irrational square roots, $\sqrt[4]{N}$ being irrational does not prove or disprove that $\sqrt{N}$ is rational.

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Example Question #302 : Arithmetic

is a positive real number. True or false: $\sqrt{N}$ is a rational number.

Statement 1: $\sqrt[3]{N}$ is a rational number.

Statement 2: $\sqrt[4]{N}$ is a rational number.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 alone provides insufficient information. is a number with a rational cube root, , and a rational square root, . is a number with a rational cube root, , but an irrational square root.

Now assume Statement 2 alone.

$\left (\sqrt[4]{N} \right )^{2}= \left (N ^{\frac{1}{4}} \right )^{2}= N ^{\frac{1}{4} \cdot 2} = N ^{\frac{1}{2} } = \sqrt{N}$

In other words, $\sqrt{N}$ is the square of $\sqrt[4]{N}$ . The rational numbers are closed under multiplication, so if $\sqrt[4]{N}$ is rational, $\left (\sqrt[4]{N} \right )^{2}= \sqrt{N}$ is rational.

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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 : Dsq: Understanding Powers And Roots

Example Question #1 : Powers & Roots Of Numbers

Example Question #2 : Dsq: Understanding Powers And Roots

Example Question #301 : Arithmetic

Example Question #4 : Dsq: Understanding Powers And Roots

Example Question #5 : Dsq: Understanding Powers And Roots

Example Question #6 : Powers & Roots Of Numbers

Example Question #6 : Dsq: Understanding Powers And Roots

Example Question #2 : Dsq: Understanding Powers And Roots

Example Question #302 : Arithmetic

All GMAT Math Resources

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