Math Section Question #57 from Official ACT Form Z18 (April 2023)

Question

Which of the following systems when solved gives the values of a and b such that (3^a)(3^b) = 9 and 2^a / 2^b = 64?

A. a+b = 2

a - b = 6

B. a + b = 9

a - b = 64

C. ab = 1

a/b = 6

D. ab = 2

a/b = 6

E. ab = 9

a/b = 64

Answer 1

By the exponent same-base rules, we can add the exponents when two numbers with the same base are multiplied: (3^a) * (3^b) = 3^a+b

We can also express 9 as 3² to get the same base on both sides: 3^a+b = 3²

since both sides have a base 3, we can focus only on the exponents and equate them. Therefore, a+b = 2

For the second half, since dividing two numbers with the same base means we can subtract the exponents, the equation becomes the following:

2^a / 2^b = 2^a-b

Applying the same reasoning as before, 64 = 2⁶

Now 2^a-b = 2⁶

Therefore a-b = 6

Answer A is correct.

Answer 2

57 Exponents and Radicals UG2 (2.14)

(3^a)(3^b) = 3^(a+b) = 9 = 3^2

a+b = 2

2^a / 2^b = 2^(a-b) = 64 = 2^6 (2*2*2*2*2*2)

a-b = 6

A