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Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{3}{x+\sqrt{2}}+\frac{5}{\sqrt{x}}$

$\frac{3(x-\sqrt{2})}{x^{2}-2}+\frac{5 \sqrt{x}}{x}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 7

Rationalizing a Denominator

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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Rationalize the denominato…

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Simplify each expression. …

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So we have three over X plus square to plus five over squared. It's so because we have these radicals in our denominator here in here, we need to get rid of those. So from the first for action, we need to multiply by its conjugated, so to congregate. In this case, we have a plus B, so it's congregate would be used in the same two terms. We're gonna change the side, so we're gonna have subtraction in the middle. This turns into a squared minus be screwed. When we finally working out and simplify, it's easier if you had memorized is conjugated formula they don't have to boil every single time. So for first fraction, we're gonna multiply the talking bottom bar the contra get someone to move this over. This would be X minus route to over X minus root, too. And our second fraction we still need to get rid of that radical. But we don't have any additional subtraction sign. So we're just gonna multiply the top and bottom by room picks. So the first fraction will have three times x minus root, too, over X squared minus root. Soon squared plus and then our second fraction will have five x over re x times re X, and I'll just give us X. It's like a simplify first fraction. Further, by distributing that three so three X minus three root, too over an X square just stays the same as good of Sue Square gives us, too. So X squared minus two and our second fraction is also simplified to its further 16.

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