Math Unit 3 Lesson 3: Exponents as Unit Fractions

Key Concepts

• Unit Fractions: Fractions with 1 in the numerator (e.g., 1/2, 1/3, 1/4).
• Exponents and Roots: Understanding the relationship between exponents and roots.
• Exponents such as ( b^{1/n} ) are equivalent to the nth root of b.

Objectives

• Write square and cube roots using exponents.
• Understand that ( b^{1/n} ) is equal to the nth root of ( b ).

Discussed Problems

Squaring and Cubing

• Squaring: When solving equations involving square roots, include both positive and negative solutions.
  • Example: ( x^2 = 25 ) has solutions ( x = \pm 5 ).
• Cubing: When solving cube roots, only the positive or negative solution is needed since cubes of negative numbers remain negative.

Exponent and Root Relations

• Square Roots as Exponents:
  • Example: ( 9^{1/2} ) is equal to the square root of 9, which is 3.
• Cube Roots as Exponents:
  • Example: ( 8^{1/3} ) is equal to the cube root of 8, which is 2.

Graphing and Estimating

• Graphing Exponential Functions: Helps in estimating fractional exponents.
  • Example: Graph ( y = 9^x ) to find approximate values for ( 9^{1/2} ).
• Use properties of exponents: ( x^m \times x^n = x^{m+n} ).

Exponent Rules

• Positive and Negative Exponents:
  • ( b^{-n} = \frac{1}{b^n} ).
• Fractional Exponents:
  • ( b^{1/2} = \sqrt{b} ).
  • ( b^{1/3} = \sqrt[3]{b} ).
  • ( b^{1/n} = \sqrt[n]{b} ).

Application and Examples

• Matching Expressions: Practice matching exponential expressions with their equivalent radical forms.
• Explaining fractional exponents and their equivalence to roots.

Summary

• Understanding Exponents: Exponents with fractions can be translated into root expressions.
• Solutions Involving Roots: Provide both positive and negative solutions when square rooting.

Practice

• Solve ( x^2 = 5 ) using both exponents and radicals: ( x = \pm \sqrt{5} ) and ( x = \pm 5^{1/2} ).
• ( 78^{1/3} ) is equivalent to the cube root of 78.

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Note: These concepts apply to any exponent that is a fraction with a unit numerator, providing a bridge between exponents and root operations.