Solve 4/sqrt{2-6}=

$$\frac { 4 } { \sqrt { 2 } - 6 } =$$

Evaluate

$$\frac{-2\sqrt{2}-12}{17}\approx -0.872260419$$

Solution Steps

Rationalize the denominator of \frac{4}{\sqrt{2}-6} by multiplying numerator and denominator by \sqrt{2}+6.

$$\frac{4\left(\sqrt{2}+6\right)}{\left(\sqrt{2}-6\right)\left(\sqrt{2}+6\right)}$$

Consider \left(\sqrt{2}-6\right)\left(\sqrt{2}+6\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4\left(\sqrt{2}+6\right)}{\left(\sqrt{2}\right)^{2}-6^{2}}$$

Square \sqrt{2}. Square 6.

$$\frac{4\left(\sqrt{2}+6\right)}{2-36}$$

Subtract 36 from 2 to get -34.

$$\frac{4\left(\sqrt{2}+6\right)}{-34}$$

Divide 4\left(\sqrt{2}+6\right) by -34 to get -\frac{2}{17}\left(\sqrt{2}+6\right).

$$-\frac{2}{17}\left(\sqrt{2}+6\right)$$

Use the distributive property to multiply -\frac{2}{17} by \sqrt{2}+6.

$$-\frac{2}{17}\sqrt{2}-\frac{2}{17}\times 6$$

Express -\frac{2}{17}\times 6 as a single fraction.

$$-\frac{2}{17}\sqrt{2}+\frac{-2\times 6}{17}$$

Multiply -2 and 6 to get -12.

$$-\frac{2}{17}\sqrt{2}+\frac{-12}{17}$$

Fraction \frac{-12}{17} can be rewritten as -\frac{12}{17} by extracting the negative sign.

$$-\frac{2}{17}\sqrt{2}-\frac{12}{17}$$