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# Algebra

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Rationalize the denominator of $\frac{1}{\sqrt{2} + \sqrt{5} + \sqrt{7}}$.

Jun 3, 2022

#1
+1

Rationalize the following:
1/(sqrt(2) + sqrt(5) + sqrt(7))

Multiply numerator and denominator of 1/(sqrt(2) + sqrt(5) + sqrt(7)) by sqrt(2) - sqrt(5) - sqrt(7):
(sqrt(2) - sqrt(5) - sqrt(7))/((sqrt(2) + sqrt(5) + sqrt(7)) (sqrt(2) - sqrt(5) - sqrt(7)

(sqrt(2) - sqrt(5) - sqrt(7))/((-2 (sqrt(35) + 5)))

((-sqrt(2) + sqrt(5) + sqrt(7)))/(2 (sqrt(35) + 5))

((-sqrt(2) + sqrt(5) + sqrt(7)) (sqrt(35) - 5))/20

((5 sqrt(2) + 2 sqrt(5) - sqrt(70)))/20

Jun 3, 2022
#2
+118622
+1

This is how I would do it.

And yes our answers do agree.  Thanks guest.

$$\frac{1}{\sqrt{2} + \sqrt{5} + \sqrt{7}}\\ \frac{1}{(\sqrt{2} + \sqrt{5}) + \sqrt{7}}*\frac{(\sqrt{2} + \sqrt{5}) - \sqrt{7}}{(\sqrt{2} + \sqrt{5}) - \sqrt{7}}\\ =\frac{(\sqrt{2} + \sqrt{5} - \sqrt{7})}{(2+5+2\sqrt{10}) -7}\\ =\frac{(\sqrt{2} + \sqrt{5} - \sqrt{7})}{(2\sqrt{10}) }\\ =\frac{(\sqrt{2} + \sqrt{5} - \sqrt{7})}{(2\sqrt{10}) }*\frac{\sqrt{10}}{\sqrt{10}}\\ =\frac{\sqrt{10}(\sqrt{2} + \sqrt{5} - \sqrt{7})}{20}\\ etc$$

You need to check my working for careless errors

Jun 4, 2022