Solve frac{sqrt{20}+sqrt{12}}{sqrt{5}-sqrt{3}}

$$\frac { \sqrt { 20 } + \sqrt { 12 } } { \sqrt { 5 } - \sqrt { 3 } }$$

Evaluate

$$2\left(\sqrt{15}+4\right)\approx 15.745966692$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{2\sqrt{5}+\sqrt{12}}{\sqrt{5}-\sqrt{3}}$$

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}.

$$\frac{2\sqrt{5}+2\sqrt{3}}{\sqrt{5}-\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{5}+2\sqrt{3}}{\sqrt{5}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}+\sqrt{3}.

$$\frac{\left(2\sqrt{5}+2\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}$$

Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{5}+2\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{3}.

$$\frac{\left(2\sqrt{5}+2\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{5-3}$$

Subtract 3 from 5 to get 2.

$$\frac{\left(2\sqrt{5}+2\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{2}$$

Apply the distributive property by multiplying each term of 2\sqrt{5}+2\sqrt{3} by each term of \sqrt{5}+\sqrt{3}.

$$\frac{2\left(\sqrt{5}\right)^{2}+2\sqrt{5}\sqrt{3}+2\sqrt{3}\sqrt{5}+2\left(\sqrt{3}\right)^{2}}{2}$$

The square of \sqrt{5} is 5.

$$\frac{2\times 5+2\sqrt{5}\sqrt{3}+2\sqrt{3}\sqrt{5}+2\left(\sqrt{3}\right)^{2}}{2}$$

Multiply 2 and 5 to get 10.

$$\frac{10+2\sqrt{5}\sqrt{3}+2\sqrt{3}\sqrt{5}+2\left(\sqrt{3}\right)^{2}}{2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{10+2\sqrt{15}+2\sqrt{3}\sqrt{5}+2\left(\sqrt{3}\right)^{2}}{2}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{10+2\sqrt{15}+2\sqrt{15}+2\left(\sqrt{3}\right)^{2}}{2}$$

Combine 2\sqrt{15} and 2\sqrt{15} to get 4\sqrt{15}.

$$\frac{10+4\sqrt{15}+2\left(\sqrt{3}\right)^{2}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{10+4\sqrt{15}+2\times 3}{2}$$

Multiply 2 and 3 to get 6.

$$\frac{10+4\sqrt{15}+6}{2}$$

Add 10 and 6 to get 16.

$$\frac{16+4\sqrt{15}}{2}$$