Solving Radicals and Simplifying Roots Lecture

Introduction

• Topic: Solving radicals and simplifying roots
• Level: Class 9th and 10th students
• Coverage: Addition, subtraction, multiplication, division of radicals
• Practical examples provided

Addition of Roots

• Similar to adding variables.
• Example 1:
  • ( \sqrt{5} + \sqrt{5} = 2\sqrt{5} )
  • Analogy: ( x + x = 2x )
• Example 2:
  • ( 2\sqrt{3} + \sqrt{3} = 3\sqrt{3} )
  • Analogy: ( 2x + x = 3x )
• Example 3:
  • ( 3\sqrt{5} + 4\sqrt{3} ) remains the same as it can't be added due to different numbers under radicals.
• Rule: Add only if the numbers under the root signs are the same.

Subtraction of Roots

• Follows similar rules as addition.
• Example 1:
  • ( \sqrt{5} - \sqrt{5} = 0 )
  • Analogy: ( x - x = 0 )
• Example 2:
  • ( 3\sqrt{2} - \sqrt{2} = 2\sqrt{2} )
  • Analogy: ( 3x - x = 2x )
• Example 3:
  • ( 4\sqrt{5} - 3\sqrt{3} ) remains the same as it can't be subtracted due to different numbers under radicals.
• Rule: Subtract only if the numbers under the root signs are the same.

Multiplication of Roots

• Using exponential representation.
  • ( \sqrt{2} = 2^{1/2} )
  • ( \sqrt{2} \times \sqrt{2} = 2^{1/2} \times 2^{1/2} = 2^{1} = 2 )
• Example 1:
  • ( \sqrt{5} \times 2\sqrt{5} = 2 \times \sqrt{5} \times \sqrt{5} = 2 \times 5 = 10 )
  • Analogy: ( x \times 2x = 2x^2 )
• Example 2:
  • ( 3\sqrt{2} \times 4\sqrt{3} = 3 \times 4 \times \sqrt{2} \times \sqrt{3} = 12\sqrt{6} )
• Rule: Multiply numbers outside the roots separately and inside the roots separately.

Division of Roots

• Using exponential rules.
  • ( \sqrt{3} = 3^{1/2} )
  • ( \sqrt{3} / \sqrt{3} = 3^{1/2} / 3^{1/2} = 3^{0} = 1 )
• Rule: Divide numbers with the same base, subtract their exponents.
• Example 1:
  • ( 2\sqrt{5} / \sqrt{5} = 2 )
• Example 2:
  • ( 3\sqrt{2} / 2\sqrt{3} ) remains the same.
• Rationalization remark: Further videos will cover rationalization of roots (making the denominator free of roots).

Solving Example Problems

• Example 1:
  • ( 3 \times \sqrt{2} + \sqrt{3} = 3\sqrt{2} + 3\sqrt{3} )
  • Rule: Numbers inside different roots can't be combined.
• Example 2:
  • ( 2\sqrt{5} \times \sqrt{5} + \sqrt{3} \times \sqrt{5} = 2 \times 5 + \sqrt{15} = 10 + \sqrt{15} )
• Example 3:
  • ( 3\sqrt{2} \times 3 - \sqrt{2} = 9\sqrt{2} - 3 \times 2 = 9\sqrt{2} - 6 )
• Example 4:
  • ( (3 + \sqrt{2}) \times (2 - \sqrt{2}) = 6 - 3\sqrt{2} + 2\sqrt{2} - 2 = 4 - \sqrt{2} )
  • Rule: FOIL Method (First, Outer, Inner, Last)

Conclusion

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• Thanks for watching.