Solve frac{8+2sqrt{2}}{4+8sqrt{2}}

$$\frac { 8 + 2 \sqrt { 2 } } { 4 + 8 \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{2}}{2}\approx 0.707106781$$

Solution Steps

Rationalize the denominator of \frac{8+2\sqrt{2}}{4+8\sqrt{2}} by multiplying numerator and denominator by 4-8\sqrt{2}.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{\left(4+8\sqrt{2}\right)\left(4-8\sqrt{2}\right)}$$

Consider \left(4+8\sqrt{2}\right)\left(4-8\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{4^{2}-\left(8\sqrt{2}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{16-\left(8\sqrt{2}\right)^{2}}$$

Expand \left(8\sqrt{2}\right)^{2}.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{16-8^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{16-64\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{16-64\times 2}$$

Multiply 64 and 2 to get 128.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{16-128}$$

Subtract 128 from 16 to get -112.

$$\frac{\left(8+2\sqrt{2}\right)\left(4-8\sqrt{2}\right)}{-112}$$

Apply the distributive property by multiplying each term of 8+2\sqrt{2} by each term of 4-8\sqrt{2}.

$$\frac{32-64\sqrt{2}+8\sqrt{2}-16\left(\sqrt{2}\right)^{2}}{-112}$$

Combine -64\sqrt{2} and 8\sqrt{2} to get -56\sqrt{2}.

$$\frac{32-56\sqrt{2}-16\left(\sqrt{2}\right)^{2}}{-112}$$

The square of \sqrt{2} is 2.

$$\frac{32-56\sqrt{2}-16\times 2}{-112}$$

Multiply -16 and 2 to get -32.

$$\frac{32-56\sqrt{2}-32}{-112}$$

Subtract 32 from 32 to get 0.

$$\frac{-56\sqrt{2}}{-112}$$

Divide -56\sqrt{2} by -112 to get \frac{1}{2}\sqrt{2}.

$$\frac{1}{2}\sqrt{2}$$