Practical examples of using conjugates to rationalize fractions.

Question

Hey there! 👋 Ever struggled with fractions that have radicals in the denominator? Using conjugates can be a lifesaver! Let's dive into how they work with some super practical examples. Ready to make those fractions rational? 😉

gregory_lin · Accepted Answer

📚 Quick Study Guide

🔢 What is a Conjugate? The conjugate of a binomial expression $a + b$ is $a - b$, and vice-versa. The key idea is that when you multiply a binomial by its conjugate, you eliminate the radical!
  💡 Why Rationalize? Rationalizing the denominator means getting rid of any radicals (like square roots) from the bottom of a fraction. This makes the fraction easier to work with.
  📝 The Process: To rationalize a fraction with a radical in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.
  ➗ Example 1: To rationalize $\frac{1}{\sqrt{2}}$, multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ to get $\frac{\sqrt{2}}{2}$.
  ➕ Example 2: To rationalize $\frac{1}{1 + \sqrt{3}}$, multiply by $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$ to get $\frac{1 - \sqrt{3}}{1 - 3} = \frac{1 - \sqrt{3}}{-2}$.

🧪 Practice Quiz

Question 1: What is the conjugate of $2 + \sqrt{5}$?
    
      $2 - \sqrt{5}$
      $-2 + \sqrt{5}$
      $-2 - \sqrt{5}$
      $2 + \sqrt{5}$

Question 2: Rationalize the denominator of $\frac{1}{\sqrt{7}}$:
    
      $\frac{\sqrt{7}}{7}$
      $\sqrt{7}$
      $\frac{7}{\sqrt{7}}$
      $\frac{1}{7}$

Question 3: Rationalize the denominator of $\frac{2}{3 - \sqrt{2}}$:
    
      $\frac{6 + 2\sqrt{2}}{7}$
      $\frac{6 - 2\sqrt{2}}{7}$
      $\frac{6 + 2\sqrt{2}}{11}$
      $\frac{6 - 2\sqrt{2}}{11}$

Question 4: Rationalize the denominator of $\frac{\sqrt{3}}{4 + \sqrt{3}}$:
    
      $\frac{4\sqrt{3} - 3}{13}$
      $\frac{4\sqrt{3} + 3}{13}$
      $\frac{-4\sqrt{3} + 3}{13}$
      $\frac{-4\sqrt{3} - 3}{13}$

Question 5: Simplify: $\frac{1 + \sqrt{2}}{1 - \sqrt{2}}$
    
      $-3 - 2\sqrt{2}$
      $3 + 2\sqrt{2}$
      $3 - 2\sqrt{2}$
      $-3 + 2\sqrt{2}$

Question 6: What should you multiply by to rationalize $\frac{5}{\sqrt{5} + \sqrt{2}}$?
    
      $\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}$
      $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}}$
      $\frac{-\sqrt{5} - \sqrt{2}}{-\sqrt{5} - \sqrt{2}}$
      $\frac{-\sqrt{5} + \sqrt{2}}{-\sqrt{5} + \sqrt{2}}$

Question 7: Rationalize the denominator of $\frac{4}{\sqrt{6} - \sqrt{2}}$:
    
      $\sqrt{6} + \sqrt{2}$
      $\sqrt{6} - \sqrt{2}$
      $2(\sqrt{6} + \sqrt{2})$
      $2(\sqrt{6} - \sqrt{2})$

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Practical examples of using conjugates to rationalize fractions.

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