Solve frac{4+sqrt{8}}{sqrt{2}-1}

$$\frac { 4 + \sqrt { 8 } } { \sqrt { 2 } - 1 }$$

Evaluate

$$6\sqrt{2}+8\approx 16.485281374$$

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}.

$$\frac{4+2\sqrt{2}}{\sqrt{2}-1}$$

Rationalize the denominator of \frac{4+2\sqrt{2}}{\sqrt{2}-1} by multiplying numerator and denominator by \sqrt{2}+1.

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

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}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 4+2\sqrt{2} by each term of \sqrt{2}+1.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Add 4 and 4 to get 8.

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

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

$$6\sqrt{2}+8$$