Rational Exponents
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Evaluate Rational Exponents
Definition:
A rational exponent is an exponent expressed as a fraction \( \frac{m}{n} \)
\[ x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m \]
Components:
• x = base (must be positive for even roots)
• m = numerator (power)
• n = denominator (root/index)
Special Cases:
Unit Fraction Exponents:
\[ x^{1/n} = \sqrt[n]{x} \]
Examples: \( 16^{1/2} = \sqrt{16} = 4 \), \( 8^{1/3} = \sqrt[3]{8} = 2 \)
Negative Rational Exponents:
\[ x^{-m/n} = \frac{1}{x^{m/n}} = \frac{1}{\sqrt[n]{x^m}} \]
Examples:
Evaluate: \( 27^{2/3} \)
Method 1: Power then root
\( 27^{2/3} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9 \)
Method 2: Root then power (easier)
\( 27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9 \)
Answer: 9
Evaluate: \( 16^{3/4} \)
\( 16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8 \)
Answer: 8
Evaluate: \( 32^{-2/5} \)
Negative exponent means reciprocal:
\( 32^{-2/5} = \frac{1}{32^{2/5}} = \frac{1}{(\sqrt[5]{32})^2} = \frac{1}{2^2} = \frac{1}{4} \)
Answer: \( \frac{1}{4} \)
Evaluate: \( 125^{4/3} \)
\( 125^{4/3} = (\sqrt[3]{125})^4 = 5^4 = 625 \)
Answer: 625
2. Convert Between Rational Exponents and Radicals
Conversion Formula:
\[ x^{m/n} \Leftrightarrow \sqrt[n]{x^m} \Leftrightarrow (\sqrt[n]{x})^m \]
Key Conversions:
• Denominator of exponent → Index of radical
• Numerator of exponent → Power inside or outside radical
• Base remains the same
Conversion Table:
| Rational Exponent | Radical Form |
|---|---|
| \( x^{1/2} \) | \( \sqrt{x} \) |
| \( x^{1/3} \) | \( \sqrt[3]{x} \) |
| \( x^{2/3} \) | \( \sqrt[3]{x^2} \) or \( (\sqrt[3]{x})^2 \) |
| \( x^{3/4} \) | \( \sqrt[4]{x^3} \) or \( (\sqrt[4]{x})^3 \) |
| \( x^{-1/2} \) | \( \frac{1}{\sqrt{x}} \) |
| \( (xy)^{2/5} \) | \( \sqrt[5]{(xy)^2} \) or \( \sqrt[5]{x^2y^2} \) |
Examples:
Convert to radical form: \( (3x)^{5/2} \)
Answer: \( \sqrt{(3x)^5} \) or \( (\sqrt{3x})^5 \)
Convert to rational exponent: \( \sqrt[4]{x^3} \)
Answer: \( x^{3/4} \)
Convert to radical form: \( x^{-3/5} \)
Answer: \( \frac{1}{\sqrt[5]{x^3}} \) or \( \frac{1}{(\sqrt[5]{x})^3} \)
3. Operations with Rational Exponents
Properties of Exponents:
Product Rule:
\[ x^{m/n} \cdot x^{p/q} = x^{(m/n) + (p/q)} \]
Add exponents when multiplying with same base
Quotient Rule:
\[ \frac{x^{m/n}}{x^{p/q}} = x^{(m/n) - (p/q)} \]
Subtract exponents when dividing with same base
Power Rule:
\[ (x^{m/n})^{p/q} = x^{(m/n) \cdot (p/q)} \]
Multiply exponents when raising a power to a power
Power of a Product:
\[ (xy)^{m/n} = x^{m/n} \cdot y^{m/n} \]
Power of a Quotient:
\[ \left(\frac{x}{y}\right)^{m/n} = \frac{x^{m/n}}{y^{m/n}} \]
Examples:
Simplify: \( x^{1/2} \cdot x^{1/3} \)
Add exponents: \( x^{1/2 + 1/3} \)
Find common denominator: \( x^{3/6 + 2/6} \)
Answer: \( x^{5/6} \)
Simplify: \( \frac{y^{3/4}}{y^{1/4}} \)
Subtract exponents: \( y^{3/4 - 1/4} \)
Answer: \( y^{2/4} = y^{1/2} \)
Simplify: \( (a^{2/3})^{3/2} \)
Multiply exponents: \( a^{(2/3) \cdot (3/2)} \)
Simplify: \( a^{6/6} \)
Answer: \( a^1 = a \)
Simplify: \( (8x^3)^{2/3} \)
Apply to each factor: \( 8^{2/3} \cdot (x^3)^{2/3} \)
Evaluate: \( (\sqrt[3]{8})^2 \cdot x^{(3)(2/3)} \)
Simplify: \( 2^2 \cdot x^2 \)
Answer: \( 4x^2 \)
4. Simplify Expressions Involving Rational Exponents
Simplification Strategies:
1. Apply exponent rules (product, quotient, power)
2. Simplify numerical bases when possible
3. Combine like terms
4. Express final answer with positive exponents
5. Reduce fractions in exponents
Examples:
Simplify: \( \frac{x^{5/6}}{x^{1/3}} \)
Subtract exponents: \( x^{5/6 - 1/3} \)
Common denominator: \( x^{5/6 - 2/6} \)
Answer: \( x^{3/6} = x^{1/2} \)
Simplify: \( (16x^4)^{3/4} \)
Distribute exponent: \( 16^{3/4} \cdot (x^4)^{3/4} \)
Evaluate: \( (\sqrt[4]{16})^3 \cdot x^{4 \cdot 3/4} \)
Simplify: \( 2^3 \cdot x^3 \)
Answer: \( 8x^3 \)
Simplify: \( \frac{27^{2/3}}{9^{1/2}} \)
Evaluate each: \( \frac{(\sqrt[3]{27})^2}{\sqrt{9}} \)
Simplify: \( \frac{3^2}{3} = \frac{9}{3} \)
Answer: 3
Simplify: \( x^{1/4} \cdot x^{3/4} \cdot x^{-1/2} \)
Add all exponents: \( x^{1/4 + 3/4 - 1/2} \)
Simplify: \( x^{4/4 - 2/4} = x^{2/4} \)
Answer: \( x^{1/2} \)
Simplify: \( \left(\frac{x^{2/3}y^{1/2}}{x^{1/6}y^{-1/4}}\right) \)
Apply quotient rule to each variable:
\( x^{2/3 - 1/6} \cdot y^{1/2 - (-1/4)} \)
\( x^{4/6 - 1/6} \cdot y^{2/4 + 1/4} \)
Answer: \( x^{1/2}y^{3/4} \)
5. Solve Equations with Rational Exponents
General Strategy:
1. Isolate the term with the rational exponent
2. Raise both sides to the reciprocal power
3. Simplify and solve for the variable
4. Check solutions (especially for even roots)
Key Technique:
To "undo" \( x^{m/n} \), raise both sides to \( n/m \)
\[ \text{If } x^{m/n} = a, \text{ then } x = a^{n/m} \]
This works because \( (x^{m/n})^{n/m} = x^{(m/n)(n/m)} = x^1 = x \)
Examples:
Solve: \( x^{2/3} = 16 \)
Raise both sides to \( 3/2 \): \( (x^{2/3})^{3/2} = 16^{3/2} \)
Simplify left: \( x^{(2/3)(3/2)} = x^1 = x \)
Simplify right: \( 16^{3/2} = (\sqrt{16})^3 = 4^3 = 64 \)
Solution: x = 64
Solve: \( x^{3/4} = 27 \)
Raise both sides to \( 4/3 \): \( (x^{3/4})^{4/3} = 27^{4/3} \)
Left side: \( x \)
Right side: \( 27^{4/3} = (\sqrt[3]{27})^4 = 3^4 = 81 \)
Solution: x = 81
Solve: \( (x + 5)^{2/3} = 4 \)
Raise both sides to \( 3/2 \): \( (x + 5)^{(2/3)(3/2)} = 4^{3/2} \)
Simplify: \( x + 5 = (\sqrt{4})^3 = 2^3 = 8 \)
Solve: \( x = 3 \)
Solution: x = 3
Solve: \( 3x^{3/2} = 24 \)
Isolate: \( x^{3/2} = 8 \)
Raise both sides to \( 2/3 \): \( x = 8^{2/3} \)
Evaluate: \( x = (\sqrt[3]{8})^2 = 2^2 = 4 \)
Solution: x = 4
Solve: \( x^{-1/2} = 5 \)
Rewrite: \( \frac{1}{x^{1/2}} = 5 \)
Cross multiply: \( 1 = 5x^{1/2} \)
Divide: \( x^{1/2} = \frac{1}{5} \)
Square both sides: \( x = \left(\frac{1}{5}\right)^2 = \frac{1}{25} \)
Solution: \( x = \frac{1}{25} \)
6. Quick Reference Summary
Essential Formulas:
Definition: \( x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m \)
Product: \( x^{m/n} \cdot x^{p/q} = x^{(m/n)+(p/q)} \)
Quotient: \( \frac{x^{m/n}}{x^{p/q}} = x^{(m/n)-(p/q)} \)
Power: \( (x^{m/n})^{p/q} = x^{(m/n)(p/q)} \)
Negative: \( x^{-m/n} = \frac{1}{x^{m/n}} \)
Solving: To undo \( x^{m/n} \), raise to \( n/m \)
📚 Study Tips
✓ Denominator of exponent = index of radical
✓ Numerator of exponent = power
✓ Usually easier to take root first, then raise to power
✓ All exponent rules apply to rational exponents
✓ Always check solutions, especially for even roots