Q2(iii):

Find : (iii) $16^{\frac{3}{4}}$

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: $16^{\frac{3}{4}} = 8$