Q2(ii):

Find : (ii) $32^{\frac{2}{5}}$

Given Expression & Initial Setup

We are required to evaluate the exponential expression:

$32^{\frac{2}{5}}$

This expression features a base of $32$ raised to a rational (fractional) exponent $\frac{2}{5}$. To simplify this, we must decompose the base into its fundamental prime factors.

Step 1: Prime Factorization of the Base

We begin by finding the prime factorization of the integer $32$. By continuously dividing by the smallest prime number ($2$), we can express $32$ as a power of $2$.

• $32 \div 2 = 16$
• $16 \div 2 = 8$
• $8 \div 2 = 4$
• $4 \div 2 = 2$
• $2 \div 2 = 1$

Thus, the prime factorization is:

$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$

Step 2: Substitution and Application of Exponent Laws

Substitute the prime factorized form ($2^5$) back into the original expression in place of $32$:

$ (2^5)^{\frac{2}{5}} $

To simplify this, we apply the Power of a Power Property of exponents [Per the fundamental law of exponents which states that $(a^m)^n = a^{m \cdot n}$]. This allows us to multiply the inner exponent by the outer rational exponent:

$ 2^{5 \times \frac{2}{5}} $

Step 3: Simplification of the Exponent

Now, perform the multiplication in the exponent. The integer $5$ in the numerator cancels out perfectly with the denominator $5$ of the fraction:

$ 5 \times \frac{2}{5} = \frac{5 \times 2}{5} = \frac{10}{5} = 2 $

Substituting the simplified exponent back onto the base of $2$ yields:

$ 2^2 $

Step 4: Final Evaluation

Evaluate the simplified exponential expression by squaring the base:

$ 2^2 = 2 \times 2 = 4 $

Final Solution: 4