Q1(ii):

Find : (ii) $32^{\frac{1}{5}}$

Initial Setup & Given Expression

We are tasked with evaluating the following exponential expression:

$32^{\frac{1}{5}}$

[Per the definition of rational exponents, an expression of the form $a^{\frac{1}{n}}$ represents the $n$-th principal root of the base $a$, denoted algebraically as $\sqrt[n]{a}$. Therefore, evaluating $32^{\frac{1}{5}}$ is mathematically equivalent to finding the 5th root of $32$.]

Step 1: Prime Factorization of the Base

To systematically simplify expressions involving fractional exponents, we first decompose the base ($32$) into its prime factors. This allows us to express the base as an exponential term, which can then be manipulated using the laws of exponents.

Dividing $32$ successively by the smallest prime number ($2$):

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

Thus, the prime factorization is $32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$.

Step 2: Substitution into the Original Expression

We substitute the prime factored form of the base back into the original expression:

$32^{\frac{1}{5}} = (2^5)^{\frac{1}{5}}$

Step 3: Application of the Laws of Exponents

[Per the Power of a Power Property of Exponents], which states that $(a^m)^n = a^{m \times n}$, we multiply the inner exponent ($5$) by the outer fractional exponent ($\frac{1}{5}$).

$(2^5)^{\frac{1}{5}} = 2^{5 \times \frac{1}{5}}$

Step 4: Simplification and Final Calculation

We now perform the multiplication in the exponent:

$5 \times \frac{1}{5} = \frac{5}{5} = 1$

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

$2^1 = 2$

Final Solution: 2