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Chapter 1: Problem 84

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

Short Answer

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

Step by step solution

01

Identify the Problem

We need to rationalize the denominator of the expression \( \frac{1}{\sqrt{x}+1} \). This involves removing the square root from the denominator.
02

Multiply by the Conjugate

To rationalize \( \frac{1}{\sqrt{x}+1} \), multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{x}-1 \). This gives us:\[\frac{1}{\sqrt{x}+1} \times \frac{\sqrt{x}-1}{\sqrt{x}-1} = \frac{\sqrt{x}-1}{(\sqrt{x}+1)(\sqrt{x}-1)}.\]
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

Simplify the Fraction

Now the expression becomes:\[\frac{\sqrt{x}-1}{x-1}.\]The denominator is now rationalized since it does not contain a square root.

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Key Concepts

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

Square Root
A square root is a value that, when multiplied by itself, gives the original number. The square root of a number \( x \) is written as \( \sqrt{x} \). For positive values, this represents the principal square root. For instance, the square root of 9 is 3 because \( 3 \times 3 = 9 \).
  • Square roots are the inverse operation of squaring numbers.
  • They are useful in solving quadratic equations and simplifying radical expressions.
In our given problem, we have \( \sqrt{x} \) as part of the denominator. This radical expression can be cumbersome in algebraic operations, hence it is necessary to remove it (or rationalize it) for ease of computation. To eliminate the square root in the denominator, we employ a technique involving the conjugate.
Difference of Squares
The difference of squares is a mathematical identity used to simplify expressions with two terms squared and subtracted. It states that:\[ (a+b)(a-b) = a^2 - b^2, \]where \( a \) and \( b \) are any expressions. This identity is instrumental in rationalizing denominators that contain square roots.
  • This technique is effective because it eliminates the middle term, \( ab - ba \).
  • It transforms expressions into simpler forms with integer coefficients.
In the problem \( \frac{1}{\sqrt{x}+1} \), after multiplying by the conjugate, \((\sqrt{x}-1)\), we apply the difference of squares to simplify the expression \((\sqrt{x}+1)(\sqrt{x}-1)\) and obtain \( x - 1 \). This transformation helps us rationalize the denominator, making it much easier to handle.
Conjugate in Algebra
In algebra, the conjugate of a binomial expression like \( a + b \) is \( a - b \), and vice versa. Using conjugates is a strategic method employed to rationalize denominators containing radicals or complex numbers.
  • Conjugates rely on the difference of squares identity to clear out radicals.
  • They are essential for rewriting expressions in a simpler form.
When handling the expression \( \frac{1}{\sqrt{x}+1} \), multiplying by the conjugate \( \sqrt{x}-1 \) helps eliminate the root by forming a difference of squares in the denominator. This results in a rational form \( \frac{\sqrt{x}-1}{x-1} \), where the denominator no longer contains a radical, simplifying further calculations.

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