Simplifying Radicals and Radical Expressions

Introduction

• Focus on square roots in this lecture.
• Learn to simplify radicals into their simplest form.
• Includes examples from simple to complex.

Types of Radical Problems

1. Smaller numbers under the root.
2. Large numbers under the root.
3. Products involving square roots.
4. Quotients and removing square roots from denominators.
5. Complex expressions requiring conjugates for simplification.

Simplifying Simple Radicals

• Perfect Squares:
  • Numbers that are the square of an integer.
  • Examples:
    • ( \sqrt{16} = 4 ) (since 4 x 4 = 16)
    • ( \sqrt{81} = 9 ) (since 9 x 9 = 81)

Simplifying Non-Perfect Squares

• Example: ( \sqrt{32} )
  • Find a perfect square that divides evenly into 32.
  • 32 = 4 x 8, rewrite as ( \sqrt{4} \times \sqrt{8} ).
  • Simplify further: ( \sqrt{4} = 2 ), ( \sqrt{8} = \sqrt{4 \times 2} ).
  • Result: ( 4 \sqrt{2} ).

Large Numbers under the Root

• Example: ( \sqrt{125} )
  • 125 = 25 x 5, rewrite as ( \sqrt{25} \times \sqrt{5} ).
  • Simplify: ( \sqrt{25} = 5 ), ( \sqrt{5} ) remains.
  • Result: ( 5 \sqrt{5} ).

Products Involving Square Roots

• Example: ( 5 \sqrt{54} )
  • Break down ( \sqrt{54} ) as ( \sqrt{9} \times \sqrt{6} ).
  • Simplify: ( \sqrt{9} = 3 ), no further simplification for ( \sqrt{6} ).
  • Result: ( 15 \sqrt{6} ).

Quotients and Removing Square Roots from Denominators

• Example: ( \frac{4}{\sqrt{2}} )
  • Multiply by ( \frac{\sqrt{2}}{\sqrt{2}} ).
  • Simplify to ( \frac{4\sqrt{2}}{2} ).
  • Result: ( 2\sqrt{2} ).

Using Conjugates

• Example: ( \frac{4}{5 - \sqrt{3}} )
  • Multiply by conjugate ( \frac{5 + \sqrt{3}}{5 + \sqrt{3}} ).
  • Simplify and combine like terms.
  • Final Result: ( \frac{23 + 9\sqrt{3}}{22} ).

Conclusion

• Simplifying radicals can make expressions easier to work with.
• Removing square roots from denominators is often required.
• Understanding the use of conjugates is essential for complex expressions.