Solve frac{sqrt{12}}{sqrt{2}-1}

$$\frac { \sqrt { 12 } } { \sqrt { 2 } - 1 }$$

Evaluate

$$2\sqrt{3}\left(\sqrt{2}+1\right)\approx 8.363081101$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{2\sqrt{3}}{\sqrt{2}-1}$$

Rationalize the denominator of \frac{2\sqrt{3}}{\sqrt{2}-1} by multiplying numerator and denominator by \sqrt{2}+1.

$$\frac{2\sqrt{3}\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}$$

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}.

$$\frac{2\sqrt{3}\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square \sqrt{2}. Square 1.

$$\frac{2\sqrt{3}\left(\sqrt{2}+1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$\frac{2\sqrt{3}\left(\sqrt{2}+1\right)}{1}$$

Anything divided by one gives itself.

$$2\sqrt{3}\left(\sqrt{2}+1\right)$$

Use the distributive property to multiply 2\sqrt{3} by \sqrt{2}+1.

$$2\sqrt{3}\sqrt{2}+2\sqrt{3}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$2\sqrt{6}+2\sqrt{3}$$