Evaluate Rational Exponents

Algebra-2

1. Fundamental Concepts

Definition: Rational exponents are expressions where the exponent is a fraction, such as $$x^{\frac{m}{n}}$$ $(n\neq0)$.
Interpretation: The expression $$x^{\frac{m}{n}}$$ can be interpreted as the nth root of $$x^m$$ or $$(\sqrt[n]{x})^m$$ $(n\neq0)$ .
Rules: When evaluating rational exponents, apply the rules of exponents and roots appropriately.

2. Key Concepts

Evaluation Rule: $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

Simplification: $$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$$

Application: Used in various mathematical contexts including geometry, physics, and engineering

3. Examples

Example 1 (Basic)

Problem: Evaluate $$8^{\frac{2}{3}}$$

Step-by-Step Solution:

Apply the rule: $$8^{\frac{2}{3}} = (\sqrt[3]{8})^2$$
Calculate the cube root: $$\sqrt[3]{8} = 2$$
Square the result: $$2^2 = 4$$

Validation: Substitute into original expression → Original: $$8^{\frac{2}{3}}$$ ; Simplified: $$4$$ ✓

Example 2 (Intermediate)

Problem: Evaluate $$16^{\frac{3}{4}}$$

Step-by-Step Solution:

Apply the rule: $$16^{\frac{3}{4}} = (\sqrt[4]{16})^3$$
Calculate the fourth root: $$\sqrt[4]{16} = 2$$
Cube the result: $$2^3 = 8$$

Validation: Substitute into original expression → Original: $$16^{\frac{3}{4}}$$ ; Simplified: $$8$$ ✓

4. Problem-Solving Techniques

Rule Application: Always start by applying the fundamental rules of rational exponents.
Root Calculation: Calculate roots before raising to powers for simplification.
Verification: Substitute values back into the original expression to verify correctness.