Lecture Notes: Rational Exponents

Review of Rules from Lesson 10-1

• Square and Cube Roots
  • If ( b^2 = a ), then ( b = \sqrt{a} )
  • ( \sqrt{a^2} = |a| )
  • If ( b^3 = a ), then ( b = \sqrt[3]{a} )
• Nth Roots
  • ( \sqrt[n]{a^n} = |a| ) when ( n ) is even
  • ( \sqrt[n]{a^n} = a ) when ( n ) is odd

Introduction to Rational Exponents

• Rational Exponents: Involve fractions
  • ( a^{1/n} ) means the nth root of ( a ) raised to the first power
  • Example: ( 64^{1/2} = \sqrt{64} = 8 )
  • Example: ( (-8)^{1/3} = \sqrt[3]{-8} = -2 )
• Expression with Variables
  • ((6x^2y)^{1/5}) is the 5th root of (6x^2y)

Rewriting Radicals with Rational Exponents

• Converting Radicals to Exponents
  • Example: ( \sqrt[5]{a} = a^{1/5} )
  • Example: ( \sqrt[7]{b} = b^{1/7} )
• Handling Non-1 Numerator
  • ( \sqrt[n]{a^m} = (\sqrt[n]{a})^m )
  • Example: ( 1000^{2/3} ) is simplified using ( (\sqrt[3]{1000})^2 )

Simplifying Expressions

• Square Root Example
  • ( \sqrt{16^3} = (\sqrt{16})^3 = 4^3 = 64 )
• Handling Negative Exponents
  • Negative exponents mean the reciprocal
  • ( a^{-n} = \frac{1}{a^n} )
  • Example: ( 100^{-1/2} = \frac{1}{\sqrt{100}} = \frac{1}{10} )

Old Rules Recap (Applicable to Rational Exponents)

• Multiplying Exponents: Add them
  • ( a^m \times a^n = a^{m+n} )
• Dividing Exponents: Subtract them
  • ( \frac{a^m}{a^n} = a^{m-n} )
• Power to a Power: Multiply the exponents
  • ( (a^m)^n = a^{mn} )

Simplifying Using Rules

• Example 1: ( 6^{1/7} \times 6^{4/7} = 6^{5/7} )
• Example 2: ( \frac{50x^{1/3}}{10x^{4/3}} = 5x^{-1} = \frac{5}{x} )

Negative Exponents in Fractions

• Example: ( 32^{-3/5} = \frac{1}{32^{3/5}} )
  • ( = \frac{1}{8} ) after simplification

Practice Problems

• Use rational exponents and simplify
• Important to distinguish between rational exponent form and radical form depending on the question

Conclusion

• Practice problems are essential for mastery
• Ensure a solid understanding before proceeding to next sections