Solve frac{sqrt{2^2+4^2}+sqrt{13}}{sqrt{20}-sqrt{13}}

$$\frac { \sqrt { 2 ^ { 2 } + 4 ^ { 2 } } + \sqrt { 13 } } { \sqrt { 20 } - \sqrt { 13 } }$$

Evaluate

$$\frac{4\sqrt{65}+33}{7}\approx 9.321290142$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$\frac{\sqrt{4+4^{2}}+\sqrt{13}}{\sqrt{20}-\sqrt{13}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\sqrt{4+16}+\sqrt{13}}{\sqrt{20}-\sqrt{13}}$$

Add 4 and 16 to get 20.

$$\frac{\sqrt{20}+\sqrt{13}}{\sqrt{20}-\sqrt{13}}$$

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}.

$$\frac{2\sqrt{5}+\sqrt{13}}{\sqrt{20}-\sqrt{13}}$$

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}.

$$\frac{2\sqrt{5}+\sqrt{13}}{2\sqrt{5}-\sqrt{13}}$$

Rationalize the denominator of \frac{2\sqrt{5}+\sqrt{13}}{2\sqrt{5}-\sqrt{13}} by multiplying numerator and denominator by 2\sqrt{5}+\sqrt{13}.

$$\frac{\left(2\sqrt{5}+\sqrt{13}\right)\left(2\sqrt{5}+\sqrt{13}\right)}{\left(2\sqrt{5}-\sqrt{13}\right)\left(2\sqrt{5}+\sqrt{13}\right)}$$

Consider \left(2\sqrt{5}-\sqrt{13}\right)\left(2\sqrt{5}+\sqrt{13}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{5}+\sqrt{13}\right)\left(2\sqrt{5}+\sqrt{13}\right)}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

Multiply 2\sqrt{5}+\sqrt{13} and 2\sqrt{5}+\sqrt{13} to get \left(2\sqrt{5}+\sqrt{13}\right)^{2}.

$$\frac{\left(2\sqrt{5}+\sqrt{13}\right)^{2}}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{5}+\sqrt{13}\right)^{2}.

$$\frac{4\left(\sqrt{5}\right)^{2}+4\sqrt{5}\sqrt{13}+\left(\sqrt{13}\right)^{2}}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{4\times 5+4\sqrt{5}\sqrt{13}+\left(\sqrt{13}\right)^{2}}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{20+4\sqrt{5}\sqrt{13}+\left(\sqrt{13}\right)^{2}}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

To multiply \sqrt{5} and \sqrt{13}, multiply the numbers under the square root.

$$\frac{20+4\sqrt{65}+\left(\sqrt{13}\right)^{2}}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{13} is 13.

$$\frac{20+4\sqrt{65}+13}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

Add 20 and 13 to get 33.

$$\frac{33+4\sqrt{65}}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

Expand \left(2\sqrt{5}\right)^{2}.

$$\frac{33+4\sqrt{65}}{2^{2}\left(\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{33+4\sqrt{65}}{4\left(\sqrt{5}\right)^{2}-\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{33+4\sqrt{65}}{4\times 5-\left(\sqrt{13}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{33+4\sqrt{65}}{20-\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{13} is 13.

$$\frac{33+4\sqrt{65}}{20-13}$$

Subtract 13 from 20 to get 7.

$$\frac{33+4\sqrt{65}}{7}$$