JAMB Mathematics 2010 Paper

Concept:Apply the rules of negative and fractional exponents, then simplify the resulting fraction.Explanation:Start with the given expression. A negative power means taking the reciprocal of the base first.$$\left( \frac{81}{16} \right)^{-\frac{1}{4}} \times 2^{-1} = \left( \frac{16}{81} \right)^{\frac{1}{4}} \times \frac{1}{2}$$Now write both numbers as fourth powers, because $16 = 2^4$ and $81 = 3^4$.$$\frac{16}{81} = \frac{2^4}{3^4} = \left( \frac{2}{3} \right)^4$$Therefore, raising to the power $\frac{1}{4}$ cancels the exponent $4$.$$\left( \frac{2}{3} \right)^{4 \times \frac{1}{4}} = \frac{2}{3}$$Multiply by the remaining factor $\frac{1}{2}$.$$\frac{2}{3} \times \frac{1}{2} = \frac{1}{3}$$This matches option A.Answer:$\frac{1}{3}$ (Option A)