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Rationalizing the Denominator by Multiplying by a Conjugate

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Rationalizing the Denominator by Multiplying by a Conjugate

Study Guide

Key Definition

Rationalizing the denominator involves eliminating radicals from the denominator. When the denominator is a binomial with a rational part and an irrational (radical) part, multiply by the conjugate. The conjugate of $a + b√x$ is $a - b√x$.

Important Notes

Multiplying by the conjugate is equivalent to creating a difference of squares.
The product of conjugates $(a + b)(a - b)$ equals $a^2 - b^2$.
Rationalizing helps in simplifying expressions.
Use the power of a product property: $(ab)^2 = a^2 b^2$.
Always simplify the final expression if possible.

Mathematical Notation

$\times$ represents multiplication

$\sqrt{x}$ represents the square root of $x$

$\frac{a}{b}$ represents a fraction

$a^2$ represents $a$ squared

Remember to use proper notation when solving problems

Why It Works

Multiplying by the conjugate eliminates the radical in the denominator by forming a difference of squares, which is a rational number.

Remember

The conjugate of $a + b$ is $a - b$. Use this to rationalize denominators.

Quick Reference

Conjugates:$(a + b)(a - b) = a^2 - b^2$

Understanding Rationalizing the Denominator by Multiplying by a Conjugate

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Video explanation of this concept

Beginner

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Beginner Explanation

To rationalize a denominator of the form $a + b\sqrt{x}$, multiply numerator and denominator by its conjugate $a - b\sqrt{x}$. This uses the identity $(a + b\sqrt{x})(a - b\sqrt{x}) = a^2 - b^2 x$, eliminating the radical from the denominator.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the conjugate of $4 + \sqrt{3}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have a fraction $\frac{3}{\sqrt{5} - 2}$ and need to rationalize its denominator. Use the conjugate to simplify the expression.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Rationalize the denominator: $\frac{2}{1 + \sqrt{3}}$.

Challenge Quiz

Single Choice Quiz

Advanced

Rationalize $\frac{5}{2 - \sqrt{3}}$.

Recap

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