Bilangan Berpangkat (1) - Bilangan Berpangkat Positif, Sifat Bilangan Berpangkat - Matematika SMP

Introduction

This tutorial covers the fundamental concepts of positive exponents in mathematics, specifically tailored for middle school students. We will explore the properties of exponents, how to perform operations with them, and the significance of these operations in solving mathematical problems.

Step 1: Understanding the Properties of Exponents

• Definition of Exponent: An exponent indicates how many times a number, called the base, is multiplied by itself. For example, (a^n) means (a) multiplied by itself (n) times.
• Key Properties:
  • Product of Powers: (a^m \times a^n = a^{m+n})
  • Quotient of Powers: ( \frac{a^m}{a^n} = a^{m-n}) (when (a \neq 0))
  • Power of a Power: ((a^m)^n = a^{m \times n})
  • Power of a Product: ((ab)^n = a^n \times b^n)
  • Power of a Quotient: (\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n})

Step 2: Exponents with Negative Bases

• When raising negative numbers to an exponent:
  • Even Exponents: The result is positive. For example, ((-2)^2 = 4).
  • Odd Exponents: The result is negative. For example, ((-2)^3 = -8).

Step 3: Multiplying and Dividing Exponents

• Multiplication:
  • When multiplying like bases, add the exponents:
    • Example: (2^3 \times 2^2 = 2^{3+2} = 2^5 = 32).
• Division:
  • When dividing like bases, subtract the exponents:
    • Example: (\frac{3^4}{3^2} = 3^{4-2} = 3^2 = 9).

Step 4: Raising Exponents to Another Exponent

• To raise an exponent to another exponent, multiply the exponents:
  • Example: ((5^2)^3 = 5^{2 \times 3} = 5^6 = 15625).

Step 5: Addition and Subtraction of Exponents

• Important Note: Exponents cannot be added or subtracted directly unless the bases are the same.
  • Example: (2^3 + 2^3 = 2 \times 2^3 = 2^4 = 16).
• If bases are different, calculate each exponent individually before adding or subtracting.

Conclusion

In this tutorial, we covered the basic properties of positive exponents, operations involving multiplication and division, and the treatment of negative bases. Understanding these concepts is crucial for solving more complex mathematical problems involving exponents. For further practice, explore problems involving different bases and operations with larger numbers.