Solve 4/sqrt{2+4}

$$\frac{ 4 }{ \sqrt{ 2 } +4 }$$

Evaluate

$$\frac{8-2\sqrt{2}}{7}\approx 0.738796125$$

Solution Steps

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

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

Consider \left(\sqrt{2}+4\right)\left(\sqrt{2}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square 4.

$$\frac{4\left(\sqrt{2}-4\right)}{2-16}$$

Subtract 16 from 2 to get -14.

$$\frac{4\left(\sqrt{2}-4\right)}{-14}$$

Divide 4\left(\sqrt{2}-4\right) by -14 to get -\frac{2}{7}\left(\sqrt{2}-4\right).

$$-\frac{2}{7}\left(\sqrt{2}-4\right)$$

Use the distributive property to multiply -\frac{2}{7} by \sqrt{2}-4.

$$-\frac{2}{7}\sqrt{2}-\frac{2}{7}\left(-4\right)$$

Express -\frac{2}{7}\left(-4\right) as a single fraction.

$$-\frac{2}{7}\sqrt{2}+\frac{-2\left(-4\right)}{7}$$

Multiply -2 and -4 to get 8.

$$-\frac{2}{7}\sqrt{2}+\frac{8}{7}$$