Solve frac{6-2sqrt{2}}{sqrt{2}-1}

$$\frac{ 6-2 \sqrt{ 2 } }{ \sqrt{ 2 } -1 }$$

Evaluate

$$4\sqrt{2}+2\approx 7.656854249$$

Solution Steps

Rationalize the denominator of \frac{6-2\sqrt{2}}{\sqrt{2}-1} by multiplying numerator and denominator by \sqrt{2}+1.

$$\frac{\left(6-2\sqrt{2}\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}$$

Consider \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(6-2\sqrt{2}\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square \sqrt{2}. Square 1.

$$\frac{\left(6-2\sqrt{2}\right)\left(\sqrt{2}+1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$\frac{\left(6-2\sqrt{2}\right)\left(\sqrt{2}+1\right)}{1}$$

Anything divided by one gives itself.

$$\left(6-2\sqrt{2}\right)\left(\sqrt{2}+1\right)$$

Apply the distributive property by multiplying each term of 6-2\sqrt{2} by each term of \sqrt{2}+1.

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

The square of \sqrt{2} is 2.

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

Multiply -2 and 2 to get -4.

$$6\sqrt{2}+6-4-2\sqrt{2}$$

Subtract 4 from 6 to get 2.

$$6\sqrt{2}+2-2\sqrt{2}$$

Combine 6\sqrt{2} and -2\sqrt{2} to get 4\sqrt{2}.

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

Factor

$$2 {(2 \sqrt{2} + 1)} = 7.656854249$$