📌 What are Rational Exponents?

A rational exponent (also called a fractional exponent) is an exponent that is a fraction.

\( a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \)

Basic Conversion Rules:

📝 Examples - Radical to Rational Exponent:

\( \sqrt{16} = 16^{1/2} = 4 \)
\( \sqrt[3]{8} = 8^{1/3} = 2 \)
\( \sqrt[4]{81} = 81^{1/4} = 3 \)
\( \sqrt{x^5} = x^{5/2} \)
\( \sqrt[3]{y^7} = y^{7/3} \)
\( (\sqrt[5]{a})^3 = a^{3/5} \)

📝 Examples - Rational Exponent to Radical:

\( 25^{1/2} = \sqrt{25} = 5 \)
\( 27^{1/3} = \sqrt[3]{27} = 3 \)
\( 32^{1/5} = \sqrt[5]{32} = 2 \)
\( x^{3/4} = \sqrt[4]{x^3} = (\sqrt[4]{x})^3 \)
\( a^{5/2} = \sqrt{a^5} = (\sqrt{a})^5 \)
\( b^{2/3} = \sqrt[3]{b^2} = (\sqrt[3]{b})^2 \)

Two Methods to Evaluate:

📝 Examples - Evaluating:

Example 1: Evaluate \( 8^{2/3} \)

Method 1 (Root first):
\( 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4 \)

Method 2 (Power first):
\( 8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4 \)

Example 2: Evaluate \( 16^{3/4} \)

\( 16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8 \)

Example 3: Evaluate \( 27^{4/3} \)

\( 27^{4/3} = (\sqrt[3]{27})^4 = 3^4 = 81 \)

Example 4: Evaluate \( 32^{3/5} \)

\( 32^{3/5} = (\sqrt[5]{32})^3 = 2^3 = 8 \)

📝 Negative Rational Exponents:

For negative rational exponents, take the reciprocal first:

\( a^{-m/n} = \frac{1}{a^{m/n}} = \frac{1}{(\sqrt[n]{a})^m} \)

Examples:

\( 4^{-1/2} = \frac{1}{4^{1/2}} = \frac{1}{\sqrt{4}} = \frac{1}{2} \)

\( 8^{-2/3} = \frac{1}{8^{2/3}} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{2^2} = \frac{1}{4} \)

\( 27^{-1/3} = \frac{1}{\sqrt[3]{27}} = \frac{1}{3} \)

All Exponent Rules Apply:

The same rules that work for integer exponents also work for rational exponents:

📝 Examples - Product Rule:

Example 1: Simplify \( x^{1/2} \cdot x^{1/3} \)

Add the exponents (find common denominator):
\( x^{1/2 + 1/3} = x^{3/6 + 2/6} = x^{5/6} \)

Example 2: Simplify \( 4^{1/2} \cdot 4^{3/2} \)

\( 4^{1/2 + 3/2} = 4^{4/2} = 4^2 = 16 \)

Example 3: Simplify \( a^{2/3} \cdot a^{1/2} \)

\( a^{2/3 + 1/2} = a^{4/6 + 3/6} = a^{7/6} \)

Example 4: Simplify \( 2x^{3/4} \cdot 5x^{1/4} \)

Multiply coefficients and add exponents:
\( 10x^{3/4 + 1/4} = 10x^{4/4} = 10x \)

📝 Examples - Quotient Rule:

Example 1: Simplify \( \frac{x^{5/6}}{x^{1/6}} \)

Subtract the exponents:
\( x^{5/6 - 1/6} = x^{4/6} = x^{2/3} \)

Example 2: Simplify \( \frac{16^{3/4}}{16^{1/4}} \)

\( 16^{3/4 - 1/4} = 16^{2/4} = 16^{1/2} = \sqrt{16} = 4 \)

Example 3: Simplify \( \frac{a^{5/2}}{a^{1/2}} \)

\( a^{5/2 - 1/2} = a^{4/2} = a^2 \)

Example 4: Simplify \( \frac{12y^{7/3}}{3y^{1/3}} \)

Divide coefficients and subtract exponents:
\( 4y^{7/3 - 1/3} = 4y^{6/3} = 4y^2 \)

📝 Examples - Power to a Power:

Example 1: Simplify \( (x^{2/3})^{3/4} \)

Multiply the exponents:
\( x^{(2/3) \cdot (3/4)} = x^{6/12} = x^{1/2} \)

Example 2: Simplify \( (4^{1/2})^3 \)

\( 4^{(1/2) \cdot 3} = 4^{3/2} = (\sqrt{4})^3 = 2^3 = 8 \)

Example 3: Simplify \( (a^{3/5})^{5/3} \)

\( a^{(3/5) \cdot (5/3)} = a^{15/15} = a^1 = a \)

Example 4: Simplify \( (8^{2/3})^{1/2} \)

\( 8^{(2/3) \cdot (1/2)} = 8^{2/6} = 8^{1/3} = \sqrt[3]{8} = 2 \)

Strategy for Simplification:

1. Apply the appropriate exponent rules
2. Combine like terms with the same base
3. Simplify fractions in exponents when possible
4. Express with positive exponents only (move negative exponents to denominator)
5. Evaluate numerical expressions when possible

📝 Examples - Simplifying (Level I):

Example 1: Simplify \( x^{1/3} \cdot x^{2/3} \cdot x^{1/2} \)

\( x^{1/3 + 2/3 + 1/2} = x^{3/3 + 1/2} = x^{1 + 1/2} = x^{3/2} \)

Example 2: Simplify \( \frac{a^{5/4}}{a^{1/4}} \cdot a^{-1/2} \)

\( a^{5/4 - 1/4 - 1/2} = a^{4/4 - 1/2} = a^{1 - 1/2} = a^{1/2} \)

Example 3: Simplify \( (x^{2/3}y^{1/2})^6 \)

Apply power to each factor:
\( x^{(2/3) \cdot 6} \cdot y^{(1/2) \cdot 6} = x^{12/3} \cdot y^{6/2} = x^4y^3 \)

📝 Examples - Simplifying (Level II):

Example 1: Simplify \( \frac{(8x^{3/4})^{2/3}}{2x^{1/6}} \)

Apply power rule to numerator:
\( \frac{8^{2/3} \cdot x^{(3/4) \cdot (2/3)}}{2x^{1/6}} = \frac{8^{2/3} \cdot x^{1/2}}{2x^{1/6}} \)

Evaluate \( 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4 \):
\( \frac{4x^{1/2}}{2x^{1/6}} = 2x^{1/2 - 1/6} = 2x^{3/6 - 1/6} = 2x^{2/6} = 2x^{1/3} \)

Example 2: Simplify \( (x^{-2/3}y^{1/2})^{-3} \)

Apply power to each factor:
\( x^{(-2/3) \cdot (-3)} \cdot y^{(1/2) \cdot (-3)} = x^{6/3} \cdot y^{-3/2} = x^2 \cdot y^{-3/2} \)

Write with positive exponents:
\( \frac{x^2}{y^{3/2}} \) or \( \frac{x^2}{\sqrt{y^3}} \)

Example 3: Simplify \( \frac{27^{1/3} \cdot 9^{1/2}}{3^{2/3}} \)

Convert to same base (base 3):
\( 27 = 3^3 \), \( 9 = 3^2 \)
\( \frac{(3^3)^{1/3} \cdot (3^2)^{1/2}}{3^{2/3}} = \frac{3^{3/3} \cdot 3^{2/2}}{3^{2/3}} = \frac{3^1 \cdot 3^1}{3^{2/3}} = \frac{3^2}{3^{2/3}} \)

\( 3^{2 - 2/3} = 3^{6/3 - 2/3} = 3^{4/3} \)

Strategy for Solving:

1. Isolate the term with the rational exponent
2. Raise both sides to the reciprocal of the exponent (this eliminates the fraction)
3. Solve the resulting equation
4. Check your solution(s) in the original equation

📝 Example 1 - Simple Equation:

Solve: \( x^{2/3} = 4 \)

Step 1: Variable term is isolated

Step 2: Raise both sides to the reciprocal power \( 3/2 \)

\( (x^{2/3})^{3/2} = 4^{3/2} \)
\( x^{(2/3) \cdot (3/2)} = 4^{3/2} \)
\( x^1 = 4^{3/2} \)
\( x = (\sqrt{4})^3 = 2^3 = 8 \)

Step 3: Check in original equation

\( 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4 \) ✓

Solution: \( x = 8 \)

📝 Example 2 - With Coefficient:

Solve: \( 3x^{3/4} = 24 \)

Step 1: Isolate the variable term

\( x^{3/4} = 8 \)

Step 2: Raise both sides to \( 4/3 \)

\( (x^{3/4})^{4/3} = 8^{4/3} \)
\( x = (\sqrt[3]{8})^4 = 2^4 = 16 \)

Step 3: Check

\( 3(16)^{3/4} = 3(\sqrt[4]{16})^3 = 3(2)^3 = 3(8) = 24 \) ✓

Solution: \( x = 16 \)

📝 Example 3 - With Addition:

Solve: \( (x - 1)^{2/3} + 3 = 7 \)

Step 1: Isolate the variable term

\( (x - 1)^{2/3} = 4 \)

Step 2: Raise both sides to \( 3/2 \)

\( [(x-1)^{2/3}]^{3/2} = 4^{3/2} \)
\( x - 1 = (\sqrt{4})^3 = 2^3 = 8 \)
\( x = 9 \)

Step 3: Check

\( (9-1)^{2/3} + 3 = 8^{2/3} + 3 = 4 + 3 = 7 \) ✓

Solution: \( x = 9 \)

📝 Example 4 - Factoring Needed:

Solve: \( 3x^{3/2} = x^{1/2} \)

Step 1: Move all terms to one side

\( 3x^{3/2} - x^{1/2} = 0 \)

Step 2: Factor out \( x^{1/2} \) (the term with lower exponent)

\( x^{1/2}(3x^{(3/2)-(1/2)} - 1) = 0 \)
\( x^{1/2}(3x^1 - 1) = 0 \)
\( x^{1/2}(3x - 1) = 0 \)

Step 3: Set each factor equal to zero

\( x^{1/2} = 0 \) → \( x = 0 \)
\( 3x - 1 = 0 \) → \( x = \frac{1}{3} \)

Solutions: \( x = 0 \) or \( x = \frac{1}{3} \)

⚠️ Important Notes:

• Always check solutions in the original equation
• When raising both sides to an even power, extraneous solutions may occur
• For even roots, the base must be non-negative
• The reciprocal of \( m/n \) is \( n/m \) (flip the fraction)
• Remember: \( (a^{m/n})^{n/m} = a^1 = a \)

⚡ Quick Summary

• Rational exponent: \( a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \)
• Numerator = power, Denominator = root (index)
• Product Rule: \( a^{m/n} \cdot a^{p/q} = a^{(m/n)+(p/q)} \) (add exponents)
• Quotient Rule: \( \frac{a^{m/n}}{a^{p/q}} = a^{(m/n)-(p/q)} \) (subtract exponents)
• Power Rule: \( (a^{m/n})^{p/q} = a^{(m/n) \cdot (p/q)} \) (multiply exponents)
• Negative exponent: \( a^{-m/n} = \frac{1}{a^{m/n}} \)
• Usually easier to take the root first, then the power
• To solve equations: raise both sides to the reciprocal of the exponent
• Always check solutions for extraneous answers
• All integer exponent rules apply to rational exponents

📚 Key Formulas Reference