Solve frac{sqrt{2}left(4-sqrt{2}right)}{2left(sqrt{2}+1right)}

$$\frac{ \sqrt{ 2 } \left( 4- \sqrt{ 2 } \right) }{ 2 \left( \sqrt{ 2 } +1 \right) }$$

Evaluate

$$5-3\sqrt{2}\approx 0.757359313$$

Solution Steps

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

$$\frac{4\sqrt{2}-\left(\sqrt{2}\right)^{2}}{2\left(\sqrt{2}+1\right)}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{2}-2}{2\left(\sqrt{2}+1\right)}$$

Use the distributive property to multiply 2 by \sqrt{2}+1.

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

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

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

Consider \left(2\sqrt{2}+2\right)\left(2\sqrt{2}-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4 from 8 to get 4.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 8 and 2 to get 16.

$$\frac{16-8\sqrt{2}-4\sqrt{2}+4}{4}$$

Combine -8\sqrt{2} and -4\sqrt{2} to get -12\sqrt{2}.

$$\frac{16-12\sqrt{2}+4}{4}$$

Add 16 and 4 to get 20.

$$\frac{20-12\sqrt{2}}{4}$$

Divide each term of 20-12\sqrt{2} by 4 to get 5-3\sqrt{2}.

$$5-3\sqrt{2}$$