McGraw-Hill Education GRE 2019

No exponent rule will work here. Instead, you must recognize the following

property: If two values are equal, then they must have the same prime factorization.

To solve for x, you should thus rewrite 8 as the product of its prime factors. 8 =

23 , so the equation now reads: 2x = 2^3. Now that the bases are equal, you know the

exponents are equal, so x = 3.

If x and y are integers, and (3x)(2y) = 324, what is the value of x + y?

SOLUTION: Rewrite 324 as the product of its prime factors:

324 = 9 × 36 = 3 × 3 × 6 × 6 = 3 × 3 × 3 × 2 × 3 × 2 = (3^4 )(2^2 ).

Thus (3x)(2y) = (3^4 )(2^2 ). Now that both sides of the equation are expressed in

terms of the same bases, you know that x = 4 and y = 2.

Factoring Exponential Expressions

Sometimes you will be given an exponent question that concerns the addition

or subtraction of exponential terms. Since exponent rules only apply to the

multiplication or division of exponents, you should almost always factor when

two exponential terms are added or subtracted. How will you do so? Look at the

following example:

232 – 2^30 is equivalent to which of the following?

A 22

B 223

C 210

D 230

E 2303

SOLUTION: 232 can be rewritten as (2^30 )(2^2 ). You can thus rewrite the expression

as (2^30 )(2^2 ) – (2^30 )(1). Since both terms share a factor of 2^30 , the expression can

be written as: 2^30 (2^2 – 1) = 2^30 (3). The correct answer is Choice E.

Roots

Roots are the opposite of exponents. Generally, a root will be denoted using the

following symbol:

If √ 16 = x, then what is x?

SOLUTION: x is the positive number that when squared yields 16. 4^2 = 16, so the

answer is 4.

274 PART 4 ■ MATH REVIEW

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