What is the simplified numerical value of 27^{\frac{2}{3}} \times 9^{-\frac{1}{2}}? — JAMB UTME

What is the simplified numerical value of $27^{\frac{2}{3}} \times 9^{-\frac{1}{2}}$ ?

Answer: 3

The simplified numerical value of the expression is 3.

Evaluating

27^{\frac{2}{3}}

gives

(\sqrt[3]{27})^2 = 3^2 = 9

. Evaluating

9^{-\frac{1}{2}}

gives

\frac{1}{\sqrt{9}} = \frac{1}{3}

. Multiplying

9

\frac{1}{3}

yields

3

Rewrite bases in terms of prime factors

27 = 3^3

and

9 = 3^2

Expressing bases in power-of-3 form allows direct application of index laws.

Simplify each indexed term using

(a^m)^n = a^{m \cdot n}

and

a^{-n} = \frac{1}{a^n}

27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9

and

9^{-\frac{1}{2}} = (3^2)^{-\frac{1}{2}} = 3^{-1} = \frac{1}{3}

Fractional indices denote roots and negative indices denote reciprocals.

Multiply the resulting numbers

9 \times \frac{1}{3} = 3

Simplifying the final product yields the single numeric answer.

Fractional and Negative Index Laws

Question