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Q2(iii):
Find :
(iii) $16^{\frac{3}{4}}$
Solution :
Given Expression & Initial Setup
We are tasked with evaluating the following exponential expression with a rational exponent:
$16^{\frac{3}{4}}$
To solve this, we must apply the fundamental laws of exponents. There are two mathematically equivalent pathways to arrive at the solution: utilizing prime factorization or converting the expression into radical notation.
Method 1: Prime Factorization and the Power of a Power Rule
- Step 1: Prime Factorization of the Base
The base of the expression is $16$. We first decompose $16$ into its prime factors.
$16 = 2 \times 2 \times 2 \times 2 = 2^4$ - Step 2: Substitution into the Original Expression
Substitute $2^4$ for $16$ in the given expression:
$16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}}$ - Step 3: Application of Exponent Laws
[Per the Power of a Power Property of Exponents: $(a^m)^n = a^{m \cdot n}$]. We multiply the inner exponent by the outer fractional exponent:
$(2^4)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}}$ - Step 4: Simplification of the Exponent
Multiply the integers in the exponent. The $4$ in the numerator and the $4$ in the denominator cancel each other out:
$4 \times \frac{3}{4} = \frac{12}{4} = 3$
The expression simplifies to:
$2^3$ - Step 5: Final Evaluation
Calculate the cube of $2$:
$2^3 = 2 \times 2 \times 2 = 8$
Method 2: Radical Notation (Fractional Exponents)
- Step 1: Conversion to Radical Form
[Per the definition of rational exponents: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$]. Here, the denominator $n=4$ represents the root index, and the numerator $m=3$ represents the power.
$16^{\frac{3}{4}} = (\sqrt[4]{16})^3$ - Step 2: Evaluate the Principal Root
Find the 4th root of $16$. We ask: what number multiplied by itself $4$ times equals $16$? Since $2^4 = 16$, the principal 4th root is $2$.
$\sqrt[4]{16} = 2$ - Step 3: Apply the Numerator Power
Substitute the evaluated root back into the expression and apply the cube power:
$(2)^3 = 8$
Visualizing the Rational Exponent Transformation
The following diagram illustrates the sequential transformation of the base $16$ through the operations dictated by the fractional exponent $\frac{3}{4}$.
Final Solution: $16^{\frac{3}{4}} = 8$
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- Q1(i): Find : (i) $64^{\frac{1}{2}}$
- Q1(ii): Find : (ii) $32^{\frac{1}{5}}$
- 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(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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