GMAT Math : DSQ: Understanding powers and roots

Example Questions

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Example Question #11 : Dsq: Understanding Powers And Roots

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

Statement 1:  is a rational number.

Statement 2:  is a rational number.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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 #11 : Powers & Roots Of Numbers

Let  be positive integers. Is  an integer?

Statement 1:  is a perfect square.

Statement 2:  is an even integer.

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.

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.

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.

Example Question #11 : Dsq: Understanding Powers And Roots

Simplify:

Explanation:

When we are faced with a radical in the denominator of a fraction, the first step is to multiply the top and bottom of the fraction by the numerator:

We can then reduce the fraction to:

Example Question #311 : Arithmetic

and  are integers. Is  positive, negative, or zero?

Statement 1:  is negative.

Statement 2:  is odd.

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.

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:

A negative integer to an even power is positive:

Example:

A negative integer to an odd power is negative:

Example:

A positive integer to an odd power is positive:

Example:

So, as seen in the first two statements, knowing only that base  is negative is insufficient to detemine the sign of ; as seen in the last two statements, knowing only that exponent  is odd is also insufficient. But by the middle statement, knowing both facts tells us  is negative.

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient.

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