Solve 5/2sqrt{7-2sqrt{5}}

$$\frac { 5 } { 2 \sqrt { 7 } - 2 \sqrt { 5 } }$$

Evaluate

$$\frac{5\left(\sqrt{5}+\sqrt{7}\right)}{4}\approx 6.102274111$$

Solution Steps

Rationalize the denominator of \frac{5}{2\sqrt{7}-2\sqrt{5}} by multiplying numerator and denominator by 2\sqrt{7}+2\sqrt{5}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{\left(2\sqrt{7}-2\sqrt{5}\right)\left(2\sqrt{7}+2\sqrt{5}\right)}$$

Consider \left(2\sqrt{7}-2\sqrt{5}\right)\left(2\sqrt{7}+2\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{\left(2\sqrt{7}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

Expand \left(2\sqrt{7}\right)^{2}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{2^{2}\left(\sqrt{7}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{4\left(\sqrt{7}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{4\times 7-\left(-2\sqrt{5}\right)^{2}}$$

Multiply 4 and 7 to get 28.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-\left(-2\sqrt{5}\right)^{2}}$$

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

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-20}$$

Subtract 20 from 28 to get 8.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{8}$$

Use the distributive property to multiply 5 by 2\sqrt{7}+2\sqrt{5}.

$$\frac{10\sqrt{7}+10\sqrt{5}}{8}$$