Math Problem Statement

5 divided by √3 - √5

Solution

To simplify the expression 535\frac{5}{\sqrt{3} - \sqrt{5}}, we need to rationalize the denominator. Here are the steps:

  1. Multiply the numerator and the denominator by the conjugate of the denominator: The conjugate of 35\sqrt{3} - \sqrt{5} is 3+5\sqrt{3} + \sqrt{5}.

5353+53+5=5(3+5)(35)(3+5)\frac{5}{\sqrt{3} - \sqrt{5}} \cdot \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \frac{5(\sqrt{3} + \sqrt{5})}{(\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5})}

  1. Simplify the denominator using the difference of squares:

(35)(3+5)=(3)2(5)2=35=2(\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2

  1. Combine the results:

5(3+5)2=5(3+5)2=532552\frac{5(\sqrt{3} + \sqrt{5})}{-2} = -\frac{5(\sqrt{3} + \sqrt{5})}{2} = -\frac{5\sqrt{3}}{2} - \frac{5\sqrt{5}}{2}

So, the simplified form of 535\frac{5}{\sqrt{3} - \sqrt{5}} is:

532552-\frac{5\sqrt{3}}{2} - \frac{5\sqrt{5}}{2}

Would you like a detailed explanation of any step? Here are 5 related questions you might find useful:

  1. How do you rationalize the denominator of a fraction?
  2. What is the difference of squares and how is it used in simplifying expressions?
  3. Can you show examples of rationalizing different types of denominators?
  4. How do you multiply and simplify expressions involving square roots?
  5. What are conjugates and why are they used in rationalizing denominators?

Tip: When rationalizing the denominator, always multiply by the conjugate of the denominator to simplify the expression effectively.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rationalization of Denominators
Conjugates
Difference of Squares

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12