EKSPONEN (BILANGAN BERPANGKAT) PECAHAN

Introduction

This tutorial will guide you through the concept of exponents, specifically focusing on fractional exponents. Understanding fractional exponents is crucial in mathematics as they represent roots and can simplify complex expressions. This guide will break down the concepts into clear steps, making it easier to grasp the fundamentals of fractional exponents.

Step 1: Understanding Exponents

• Definition of an Exponent: An exponent represents how many times a number (the base) is multiplied by itself.
• Examples:
  • (2^3 = 2 \times 2 \times 2 = 8)
  • Here, 2 is the base and 3 is the exponent.

Step 2: What are Fractional Exponents?

• Definition: A fractional exponent indicates both a power and a root.
• General Form:
  • The expression (a^{m/n}) means the nth root of (a) raised to the mth power.
  • For example:
    • (a^{1/2} = \sqrt{a}) (square root)
    • (a^{1/3} = \sqrt[3]{a}) (cube root)

Step 3: Converting Fractional Exponents to Roots

• Conversion Process:
  • Identify the numerator and denominator of the fractional exponent.
  • The numerator indicates the power, while the denominator indicates the root.
• Example:
  • For (x^{3/4}):
    • Take the fourth root of (x) and then raise it to the third power: [ x^{3/4} = \sqrt[4]{x^3} ]

Step 4: Simplifying Expressions with Fractional Exponents

• Step-by-Step Simplification:
  1. Rewrite the expression using fractional exponents.
  2. Apply the rules of exponents, such as multiplying and dividing bases.
  3. Simplify the expression as needed.
• Example:
  • Simplifying (16^{3/4}):
    • First, recognize that (16 = 2^4).
    • Rewrite as ((2^4)^{3/4}).
    • Apply the power of a power rule: (2^{4 \times (3/4)} = 2^3 = 8).

Step 5: Practice Problems

• Problem 1: Simplify (27^{2/3}).

  • Solution: (27 = 3^3), thus (27^{2/3} = (3^3)^{2/3} = 3^{3 \times (2/3)} = 3^2 = 9).

• Problem 2: Simplify (64^{1/2}).

  • Solution: (64 = 8^2), thus (64^{1/2} = \sqrt{64} = 8).

Conclusion

Understanding fractional exponents is essential for simplifying mathematical expressions and solving equations. By mastering the conversion of fractional exponents to roots and applying the rules of exponents, you can tackle a wide range of math problems. For further practice, try simplifying various expressions using the methods outlined in this tutorial.