Answered: Rationalize the numerator : [√(x +5) -…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

ChapterP: Prerequisites

SectionP.7: Rational Expressions

Problem 94E

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Question

Rationalize the numerator : [√(x +5) - √x] / 5

Expert Solution

Step 1

Given- $\frac{[\sqrt{(x + 5)} - \sqrt{x}]}{5}$

To Rationlise- The numerator.

Concept Used- Ratiolisation of numerator means we have to multiply by conjugate of the numerator in both numerator and denominator.

Formula used- $(a + b) (a - b) = a^{2} - b^{2}$

Step 2

Explanation- Rewrite the given expression,

$\frac{[\sqrt{(x + 5)} - \sqrt{x}]}{5}$

As we have to rationlise the numerator , so we have to multiply by conjugate of the $[\sqrt{(x + 5)} - \sqrt{x}]$ is $[\sqrt{(x + 5)} + \sqrt{x}]$ to numerator and denominator both.

So, the above expression becomes,

$\frac{[\sqrt{(x + 5)} - \sqrt{x}]}{5} = \frac{\sqrt{(x + 5)} - \sqrt{x}}{5} \cdot \frac{\sqrt{(x + 5)} + \sqrt{x}}{\sqrt{(x + 5)} + \sqrt{x}} = \frac{{(\sqrt{x + 5})}^{2} - {(\sqrt{x})}^{2}}{5 \cdot [\sqrt{(x + 5)} + \sqrt{x}]} = \frac{x + 5 - x}{5 [\sqrt{(x + 5)} + \sqrt{x}]} = \frac{5}{5 [\sqrt{(x + 5)} + \sqrt{x}]} = \frac{1}{[\sqrt{(x + 5)} - \sqrt{x}]}$

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