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Introduction

This tutorial will guide you through the process of rationalizing denominators that involve square roots. This technique is crucial for simplifying fractions in algebra and is particularly useful in various mathematical applications. By the end of this guide, you'll understand how to manipulate expressions with square roots in their denominators effectively.

Step 1: Identify the Denominator

• Look for expressions that contain square roots in the denominator.
• For example, in the fraction ( \frac{1}{\sqrt{5}} ), the denominator is ( \sqrt{5} ).

Step 2: Multiply by the Conjugate

• To rationalize the denominator, you will multiply both the numerator and denominator by the square root present in the denominator.
• Using the previous example:
  • Multiply ( \frac{1}{\sqrt{5}} ) by ( \frac{\sqrt{5}}{\sqrt{5}} ).
  • This results in:
    [ \frac{1 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{\sqrt{5}}{5} ]

Step 3: Simplify the Expression

• After performing the multiplication, simplify the fraction if possible.
• In our example, ( \frac{\sqrt{5}}{5} ) is already in its simplest form.

Step 4: Handling Complex Denominators

• For denominators that are sums or differences involving square roots (e.g., ( \sqrt{a} + \sqrt{b} )), multiply by the conjugate:
  • The conjugate of ( \sqrt{a} + \sqrt{b} ) is ( \sqrt{a} - \sqrt{b} ).
• For example:
  • Starting with ( \frac{1}{\sqrt{2} + \sqrt{3}} ), multiply by ( \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} ): [ \frac{1 \cdot (\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})} ]
• The denominator simplifies using the difference of squares: [ (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1 ]
• Thus, the expression becomes: [ \frac{\sqrt{2} - \sqrt{3}}{-1} = -(\sqrt{2} - \sqrt{3}) = \sqrt{3} - \sqrt{2} ]

Step 5: Practice with Different Examples

• Try rationalizing the following denominators:
  • ( \frac{1}{\sqrt{7}} )
  • ( \frac{2}{\sqrt{4} + \sqrt{5}} )
  • ( \frac{5}{\sqrt{8} - \sqrt{2}} )
• Follow the same steps: identify the denominator, multiply by the conjugate, and simplify.

Conclusion

Rationalizing denominators involving square roots is a fundamental skill in algebra. By multiplying by the appropriate conjugate, you can simplify expressions and make calculations easier. Practice with various examples to reinforce your understanding and improve your skills. For further learning, consider tackling more complex expressions or exploring additional algebraic techniques.