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# What is the simplest form of the radical expression of (sqrt2+sqrt5)/(sqrt2-sqrt5)?

Apr 23, 2015

Multiply and divide by $\sqrt{2} + \sqrt{5}$ to get:
${\left[\sqrt{2} + \sqrt{5}\right]}^{2} / \left(2 - 5\right) = - \frac{1}{3} \left[2 + 2 \sqrt{10} + 5\right] = - \frac{1}{3} \left[7 + 2 \sqrt{10}\right]$

Feb 5, 2017

Conjugate

#### Explanation:

We decided to multiply the top and the bottom by $\sqrt{2} + \sqrt{5}$ because this is the conjugate of the denominator, $\sqrt{2} - \sqrt{5}$.

A conjugate is an expression in which the sign in the middle is reversed. If (A+B) is the denominator, then (A-B) would be the conjugate expression.

When simplifying square roots in the denominators, try multiplying the top and bottom by the conjugate. It will get rid of the square root, because $\left(A + B\right) \left(A - B\right) = {A}^{2} - {B}^{2}$, meaning you will be left with the numbers in the denominator squared.