Solve 2(1/sqrt{2-1})

$$2( \frac{ 1 }{ \sqrt{ 2 } -1 } )$$

Evaluate

$$2\sqrt{2}+2\approx 4.828427125$$

Solution Steps

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

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

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}.

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

Square \sqrt{2}. Square 1.

$$2\times \frac{\sqrt{2}+1}{2-1}$$

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

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

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