Solve sqrt{8}div(sqrt{2}-1)

$$\sqrt { 8 } \div ( \sqrt { 2 } - 1 )$$

Evaluate

$$2\sqrt{2}+4\approx 6.828427125$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

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}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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