Simplifying Fractional Exponents

Introduction

• Simplifying fractional exponents involves separating the fraction into two parts.
• Example: to simplify (8^{2/3}), break it into (8^{1/3}) raised to the 2nd power.

Steps to Simplify

1. Find the Root:
   • For (8^{1/3}), find the cube root of 8.
   • Cube root of 8 is 2 because (2 \times 2 \times 2 = 8).
2. Raise to the Power:
   • (2^2 = 4).
   • Therefore, (8^{2/3} = 4).

Examples

• Example 1: (16^{5/4})
  • 4th root of 16 is 2.
  • (2^5 = 32).
• Example 2: (9^{3/2})
  • Square root of 9 is 3.
  • (3^3 = 27).
• Example 3: Cube root of 64 squared:
  • Cube root of 64 is 4.
  • (4^2 = 16).

Handling Negative Exponents

• Example: (25^{-3/2})
  • Convert negative exponent: (\frac{1}{25^{3/2}}).
  • Square root of 25 is 5.
  • (5^3 = 125).
  • Result: (\frac{1}{125}).

Simplifying Expressions with Different Roots

• Example: Cube root of (x^4) times fourth root of (x^5).
  • Convert to fractional exponents: (x^{4/3}) and (x^{5/4}).
  • Add exponents using common denominators.
  • Result: (x^{31/12}), or 12th root of (x^{31}).
  • Simplify to (x^2 \times \text{12th root of } x^7).

Simplifying Division of Roots

• Example: Fifth root of (x^3) divided by fourth root of (x^7).
  • Convert to fractional exponents: (x^{3/5}) and (x^{7/4}).
  • Subtract exponents using common denominators.
  • Result: (x^{-23/20}), or (\frac{1}{x^{23/20}}).
  • Simplify to absolute value form, if needed.

Conclusion

• Practice simplifying fractional exponents using these methods.
• For more resources, check additional topics on algebra and other subjects.