How do you simplify (2sqrt 8 + 7sqrt8)/(1 - sqrt2)?

Apr 15, 2015

We can add the terms in the numerator and then rationalize:
$\frac{9 \sqrt{8}}{1 - \sqrt{2}} \cdot \frac{1 + \sqrt{2}}{1 + \sqrt{2}} =$
$= \frac{9 \sqrt{8} \left(1 + \sqrt{2}\right)}{1 - 2} =$
$= - 9 \sqrt{4 \cdot 2} \left(1 + \sqrt{2}\right) =$
$= - 18 \sqrt{2} \left(1 + \sqrt{2}\right)$

Apr 15, 2015
1. Simplify the numerator.
2. Multiply the numerator and denominator by the conjugate of the denominator.
3. Simplify a bit more.

Step 1
$2 \sqrt{8} + 7 \sqrt{8}$
$= 9 \sqrt{8}$
$= 9 \sqrt{{2}^{\cdot} 2}$
$= 18 \sqrt{2}$

Step2
$\frac{18 \sqrt{2}}{1 - \sqrt{2}} \cdot \frac{1 + \sqrt{2}}{1 + \sqrt{2}}$

= (18sqrt(2)+36)/(1-(sqrt(2)^2)

Step 3
$= - \left(36 + 18 \sqrt{2}\right)$