GMAT Math : Arithmetic

Example Questions

Example Question #301 : Arithmetic

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

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 . There is not enough time on the test for you to try to do this by hand.

Example Question #5 : Powers & Roots Of Numbers

Calculate the fifth root of :

(1) The square root of  is .

(2) The tenth root of  is .

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 not sufficient.

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

Each statement ALONE is sufficient.

Each statement ALONE is sufficient.

Explanation:

Using Statement (1):

Statement (1) ALONE is SUFFICIENT.

Using Statement (2):

Statement (2) ALONE is SUFFICIENT.

Therefore EACH  Statement ALONE is sufficient.

Example Question #6 : Powers & Roots Of Numbers

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

Statement 1:  is an irrational number.

Statement 2:  is an irrational number.

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

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.

Example Question #7 : Powers & Roots Of Numbers

. True or false:  is rational.

Statement 1:  is rational.

Statement 2:  is rational.

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.

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.

Explanation:

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

If , then

If , then

In both cases,  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.  and  are rational, so their difference is as well:

is rational, so by closure under division,  is rational.

Example Question #8 : Powers & Roots Of Numbers

. True or false:  is rational.

Statement 1:  is irrational.

Statement 2:  is rational.

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.

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.

Explanation:

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

If , then

If  , then

In both cases,  is irrational, but in only one case,  is rational.

Assume Statement 2 alone.  is rational, so, by closure of the rational numbers under multiplication,

is rational. The rationals are closed under addition, so the sum

is rational.

Example Question #9 : Powers & Roots Of Numbers

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

Statement 1:  is irrational.

Statement 2:  is irrational.

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.

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.

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.

Explanation:

If  is rational, then, since the product of two rational numbers is rational,  is rational. If Statement 1 alone is assumed, then, since  is irrational,  must be irrational.

Assume Statement 2 alone, and note that

In other words,  is the square root of . Since both rational and irrational numbers have irrational square roots,  being irrational does not prove or disprove that  is rational.

Example Question #10 : Powers & Roots Of Numbers

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

Statement 1:  is a rational number.

Statement 2:  is a rational number.

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.

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.

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

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.

In other words, is the square of . The rational numbers are closed under multiplication, so if  is rational,   is rational.

Example Question #11 : Powers & Roots Of Numbers

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

Statement 1:  is a rational number.

Statement 2:  is a rational number.

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 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.

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 determine whether  is rational or not;  and  both have rational cubes, but only  is rational. By a similar argument, Statement 2 alone is insufficient.

Assume both statements are true. , the quotient of two rational numbers, which must itself be rational.

Example Question #12 : Powers & Roots Of Numbers

Let  be positive integers. Is  an integer?

Statement 1:  is a perfect square.

Statement 2:  is an even integer.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We examine two examples of situations in which both statements hold.

Example 1:

Then

32 is not a perfect square, so  is not an integer.

Example 2:

Then , making   an integer.

In both cases, both statements hold, but in only one,  is an integer. This makes the two statements together insufficient.

Simplify: