Free GMAT Problem Solving Practice Question

Method (rewrite every term over a common base of 2). Notice that 4, 2, and 8 are all powers of 2, so convert each term before combining. Write 4³ as (2²)³ = 2⁶ (multiply the exponents: 2 × 3 = 6), and write 8⁴ as (2³)⁴ = 2¹² (multiply the exponents: 3 × 4 = 12). The numerator is then 2⁶ × 2⁵ = 2¹¹ (add the exponents when multiplying the same base). Dividing by the denominator subtracts exponents: 2¹¹ divided by 2¹² = 2⁽¹¹⁻¹²⁾ = 2⁻¹. A base raised to a negative exponent is the reciprocal of the positive power, so 2⁻¹ = 1/2. The correct answer is (C).

Fast check (no large numbers): you never need to compute 4³ = 64 or 8⁴. Once every term is base 2, the whole problem is bookkeeping on exponents: 6 + 5 − 12 = −1, then read 2⁻¹ as 1/2.

Why each wrong choice is tempting. (A) 1/16 comes from base confusion: keeping the visible exponent 3 on 4³ as though 4 were already base 2, instead of rewriting 4 as 2². (B) 1/4 comes from converting 4³ with addition (2 + 3 = 5) rather than multiplication (2 × 3 = 6), an error in the power of a power rule. (E) 16 comes from the same addition slip on the denominator, taking 8⁴ as 2⁷ instead of 2¹². (D) 2 is the strongest trap: it uses every rule correctly to reach a net exponent of −1, then misreads 2⁻¹ as 2 rather than as 1/2, dropping the negative sign and reporting the reciprocal of the true answer.