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Chapter 1: Problem 88

Rationalize the denominator. $$\frac{1}{\sqrt{x}+1}$$

Short Answer

Expert verified
The rationalized form is \( \frac{\sqrt{x} - 1}{x - 1} \).

Step by step solution

01

Identify the Conjugate

The denominator is \( \sqrt{x} + 1 \). The conjugate of this expression is \( \sqrt{x} - 1 \).
02

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( \sqrt{x} - 1 \) to rationalize the denominator. This gives: \[ \frac{1}{\sqrt{x} + 1} \cdot \frac{\sqrt{x} - 1}{\sqrt{x} - 1} = \frac{\sqrt{x} - 1}{(\sqrt{x} + 1)(\sqrt{x} - 1)}. \]
03

Simplify the Denominator

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to simplify the denominator: \( (\sqrt{x}+1)(\sqrt{x}-1) = \sqrt{x}^2 - 1^2 = x - 1 \).
04

Write the Final Expression

The expression now is \( \frac{\sqrt{x} - 1}{x - 1} \). The denominator is rationalized.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjugate
When dealing with irrational numbers in a denominator, a powerful technique is to use the "conjugate." The conjugate of a two-term expression of the form \( a + b \) or \( a - b \) is simply the same expression but with the opposite sign between the terms. This means:
  • The conjugate of \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \).
  • Similarly, the conjugate of \( a - b \) would be \( a + b \).
This concept is especially useful because multiplying an expression by its conjugate eliminates the square root in the denominator, making it easier to simplify and work with. By turning a square root into a perfect square using its conjugate, we prepare it for further simplification.
Difference of Squares
The difference of squares is a well-known algebraic identity expressed as \( (a+b)(a-b) = a^2 - b^2 \). It is particularly useful in simplifying expressions involving conjugates. Consider the example in the original exercise:
  • The denominator is \( (\sqrt{x} + 1)(\sqrt{x} - 1) \).
  • Using the difference of squares formula, this becomes \( (\sqrt{x})^2 - 1^2 \).
  • Simplifying further, it results in \( x - 1 \).
Employing the difference of squares allows us to cancel out irrational components by transforming them into a solely rational expression. It's an essential tool to keep our calculations tidy, especially when rationalizing denominators.
Simplifying Expressions
Simplifying expressions involves reducing them to their most basic form. This makes them easier to read and analyze. In the context of rationalizing denominators, simplifying means:
  • Using algebraic identities, like the difference of squares, to rewrite parts of the expression.
  • Eliminating irrational numbers from denominators by multiplying with conjugates.
In the exercise example, we started with the expression \( \frac{1}{\sqrt{x} + 1} \). By multiplying both the numerator and denominator by the conjugate \( \sqrt{x} - 1 \), and then applying the difference of squares, the denominator becomes a simpler \( x - 1 \). This step is critical because it turns the expression into \( \frac{\sqrt{x} - 1}{x - 1} \), free from square roots in the denominator. Always remember, simplifying is about clarity and making a problem more approachable.

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Most popular questions from this chapter

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