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Q1(ii):
Find :
(ii) $32^{\frac{1}{5}}$
Solution :
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
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- Q1(i): Find : (i) $64^{\frac{1}{2}}$
- Q1(iii): Find : (iii) $125^{\frac{1}{3}}$
- Q2(i): Find : (i) $9^{\frac{3}{2}}$
- Q2(ii): Find : (ii) $32^{\frac{2}{5}}$
- Q2(iii): Find : (iii) $16^{\frac{3}{4}}$
- Q2(iv): Find : (iv) $125^{-\frac{1}{3}}$
- Q3(i): Simplify : (i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
- Q3(ii): Simplify : (ii) $(\frac{1}{3^3})^7$
- Q3(iii): Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
- Q3(iv): Simplify : (iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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