GMAT Math : Powers & Roots of Numbers

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Example Question #1 :Dsq: Understanding Powers And Roots

is a real number. Ispositive, negative, or zero?

Statement 1: 2x - 5 > 0

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

Possible Answers:

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.

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.

EITHER statement ALONE is 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 $x > \frac{5}{2} > 0$ , somustbe 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,mustbe 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:

$3 \sqrt[3]{2}$

$\sqrt[3]{198}$

The expression is already simplified

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

$9 \sqrt[3]{2}$

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 #3 :Powers & Roots Of Numbers

Imagine an integersuch that the units digit ofis greater than 5. What is the units digit of?

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

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

Possible Answers:

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

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 #1 :Powers & Roots Of Numbers

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

Possible Answers:

1,728

28,380

144

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 #5 :Powers & Roots Of Numbers

Calculate the fifth root of:

(1) The square root ofis 3125 .

(2) The tenth root ofis.

Possible Answers:

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

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

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

Both statements TOGETHER are not sufficient.

Each 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 #6 :Powers & Roots Of Numbers

is a positive real number. True or false:是一个理性的number.

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

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

Possible Answers:

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

一个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 provesirrational.

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Example Question #7 :Powers & Roots Of Numbers

N > 1 . True or false:是理性的。

Statement 1: $\left (N + 1 \right )^{2}$ 是理性的。

Statement 2: $\left (N - 1 \right )^{2}$ 是理性的。

Possible Answers:

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

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

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

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 proveis 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, $4N \div 4 = N$ 是理性的。

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Example Question #1 :Powers & Roots Of Numbers

N > 1 . True or false:是理性的。

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

Statement 2: $\sqrt{N-1}$ 是理性的。

Possible Answers:

EITHER 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.

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.

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 proverational 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,是理性的。

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

$\sqrt{N-1} \cdot \sqrt{N-1} = N - 1$ 是理性的。The rationals are closed under addition, so the sum

N - 1+ 1 = N

是理性的。

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Example Question #9 :Powers & Roots Of Numbers

is a positive real number. True or false: $\sqrt{N}$ 是一个理性的number.

Statement 1:is irrational.

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

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 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, $\sqrt{N} \cdot \sqrt{N} = N$ 是理性的。If Statement 1 alone is assumed, then, sinceis 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 $\sqrt{N}$ 是理性的。

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Example Question #10 :Powers & Roots Of Numbers

is a positive real number. True or false: $\sqrt{N}$ 是一个理性的number.

Statement 1: $\sqrt[3]{N}$ 是一个理性的number.

Statement 2: $\sqrt[4]{N}$ 是一个理性的number.

Possible Answers:

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.

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.

EITHER statement ALONE is 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, $\left (\sqrt[4]{N} \right )^{2}= \sqrt{N}$ 是理性的。

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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 #3 :Powers & Roots Of Numbers

Example Question #1 :Powers & Roots Of Numbers

Example Question #5 :Powers & Roots Of Numbers

Example Question #6 :Powers & Roots Of Numbers

Example Question #7 :Powers & Roots Of Numbers

Example Question #1 :Powers & Roots Of Numbers

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