Lecture Notes: Nth Roots and Rational Exponents

Introduction

• Topic: Understanding nth roots and rational exponents.
• Importance: Recognizing powers like 125 = 5³, 27 = 3³, 81 = 3⁴ is crucial.

Rational Exponents

• Definition: A fraction in the exponent (e.g., a^(1/n)).
  • Example: a^(1/2) is the square root of a.
• Conversion:
  • Rational exponent form: a^(1/n).
  • Radical form: nth root of a.

Index and Radical Expressions

• Index (n): In a^(1/n), n is the index.
  • Example: a^(1/3) is the cube root of a, index = 3.
• Rewriting:
  • Convert between radical and rational exponent forms.

Examples with Rational Exponents

• Square Root:
  • x^(1/2) = √x.
• Cube Root:
  • x^(1/3) = ³√x.
• Examples:
  • √16 = 4 because 4² = 16.
  • ³√27 = 3 because 3³ = 27.
  • ⁴√625 = 5 because 5⁴ = 625.
  • ⁴√16 = 2 because 2⁴ = 16.

Rewriting and Evaluating

• Examples:
  • 8^(1/3) = ³√8 = 2.
  • (-27)^(1/3) = ³√(-27) = -3.
  • 81^(1/4) = ⁴√81 = 3.
  • (-64)^(1/3) = ³√(-64) = -4.
  • 64^(1/6) = ⁶√64 = 2.

Rational Exponents with Non-Unit Numerators

• Example: 16^(3/2)
  • Rewrite: 16^(1/2) * 3.
  • Solve: 4³ = 64.
• Example: 27^(2/3)
  • Rewrite: 27^(1/3) * 2.
  • Solve: 3² = 9.

Negative Exponents

• Example: 32^(-3/5)
  • Rewrite: 1/32^(3/5).
  • Solve and simplify.
• Example: 81^(-3/4)
  • Solve similarly to above.

Solving Equations with Rational Exponents

• Method:
  • Isolate the variable.
  • Take the nth root based on the exponent.
• Examples:
  • 4x⁵ = 128, solve for x: x = 2.
  • 3x⁴ = 48, solve for x: x = ±2.

Complex Examples

• Equations with Parentheses
  • E.g., solve 3(x+5)³ = 81.
  • Isolate the parentheses, take the cube root, solve.

Conclusion

• Understanding and converting between rational exponents and radical forms is essential.
• Being able to solve equations with higher powers requires mastery of these concepts.