GMAT Practice Question: If a and b are positive integers and (2^{a})^{b}=2^{3}, what is the value of 2^{a} 2^{b} ?
Question
If a and b are positive integers and (2^{a})^{b}=2^{3}, what is the value of 2^{a} 2^{b} ?
- 6
- 8
- 16
- 32
- 64
Topics: exponents, calculations
Solution
Step 1
We rewrite the left side using exponent rules and then set the exponents equal because the base is 2 on both sides.
(2^a)^b = 2^3 2^{a b} = 2^3 a b = 3
Step 2
We list the positive integer pairs that multiply to 3.
a \times b = 3 (a,b) = (1,3) \text{ or } (3,1)
Step 3
We add the exponents to get 4 and then evaluate the corresponding power of two.
2^a \times 2^b = 2^{a + b} = 2^4 = 16
Answer
16
