Eksponen & Logaritma Bagian 3 - Bentuk Akar - Kelas X Kurikulum Merdeka

Introduction

This tutorial covers the key concepts of exponents and logarithms, specifically focusing on roots as discussed in the video "Eksponen & Logaritma Bagian 3 - Bentuk Akar" by m4th-lab. Understanding these mathematical principles is essential for students, particularly in Kelas X under the Kurikulum Merdeka. We will explore the relationship between roots and rational exponents, perform algebraic operations with roots, and learn how to rationalize denominators.

Step 1: Understand the Relationship Between Roots and Rational Exponents

• Concept of Roots: A root can be expressed as a power. For example, the square root of a number can be represented as that number raised to the power of 1/2.
• Rational Exponents:
  • The nth root of a number ( a ) is expressed as ( a^{1/n} ).
  • Example:
    • ( \sqrt{a} = a^{1/2} )
    • ( \sqrt[3]{a} = a^{1/3} )

Step 2: Perform Algebraic Operations with Roots

Addition of Roots

1. Like Terms: Only add roots with the same radicand (the number inside the root).
   • Example:
     • ( \sqrt{2} + \sqrt{2} = 2\sqrt{2} )
     • ( \sqrt{3} + \sqrt{2} ) cannot be simplified further.

Multiplication of Roots

1. Multiplying Roots: Multiply the radicands.
   • Example:
     • ( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} )

Step 3: Rationalize the Denominator

• Rationalizing: To eliminate roots from the denominator of a fraction.
• Steps to Rationalize:
  1. Identify the root in the denominator.
  2. Multiply the numerator and denominator by the same root.
  • Example:
    • To rationalize ( \frac{1}{\sqrt{a}} ):
      • Multiply by ( \frac{\sqrt{a}}{\sqrt{a}} ) to get ( \frac{\sqrt{a}}{a} ).

Step 4: Practice with Examples

• Example 1: Simplify ( \sqrt{8} + \sqrt{2} ).

  • Solution:
    • Rewrite ( \sqrt{8} ) as ( 2\sqrt{2} ).
    • Combine: ( 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} ).

• Example 2: Rationalize ( \frac{5}{\sqrt{3}} ).

  • Solution:
    • Multiply by ( \frac{\sqrt{3}}{\sqrt{3}} ) to get ( \frac{5\sqrt{3}}{3} ).

Conclusion

In this tutorial, we explored the connections between roots and rational exponents, performed addition and multiplication of roots, and learned how to rationalize denominators. These skills are foundational for further studies in algebra and mathematics. To enhance your understanding, practice more problems involving roots and consider joining live classes for interactive learning.