Question

question
express as a fraction in simplest form with a rational denominator:
$\frac{-2}{8+sqrt{2}}$

Explanation:

Step1: Rationalize denominator with conjugate

Multiply numerator and denominator by $8-\sqrt{2}$:
$\frac{-2}{8+\sqrt{2}} \times \frac{8-\sqrt{2}}{8-\sqrt{2}} = \frac{-2(8-\sqrt{2})}{(8+\sqrt{2})(8-\sqrt{2})}$

Step2: Expand denominator via difference of squares

Use $(a+b)(a-b)=a^2-b^2$:
Denominator: $8^2 - (\sqrt{2})^2 = 64 - 2 = 62$

Step3: Expand numerator

Distribute $-2$ across terms:
Numerator: $-2\times8 + (-2)\times(-\sqrt{2}) = -16 + 2\sqrt{2}$

Step4: Simplify the fraction

Divide numerator terms by 62:
$\frac{-16 + 2\sqrt{2}}{62} = \frac{-8 + \sqrt{2}}{31}$

Answer:

$\frac{-8 + \sqrt{2}}{31}$ (or equivalently $\frac{\sqrt{2}-8}{31}$)