Solve 1/7sqrt{5+5sqrt{7}}

$$\frac { 1 } { 7 \sqrt { 5 } + 5 \sqrt { 7 } }$$

Evaluate

$$\frac{\sqrt{7}\left(\sqrt{35}-5\right)}{70}\approx 0.034624561$$

Solution Steps

Rationalize the denominator of \frac{1}{7\sqrt{5}+5\sqrt{7}} by multiplying numerator and denominator by 7\sqrt{5}-5\sqrt{7}.

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

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

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

Expand \left(7\sqrt{5}\right)^{2}.

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

Calculate 7 to the power of 2 and get 49.

$$\frac{7\sqrt{5}-5\sqrt{7}}{49\left(\sqrt{5}\right)^{2}-\left(5\sqrt{7}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{7\sqrt{5}-5\sqrt{7}}{49\times 5-\left(5\sqrt{7}\right)^{2}}$$

Multiply 49 and 5 to get 245.

$$\frac{7\sqrt{5}-5\sqrt{7}}{245-\left(5\sqrt{7}\right)^{2}}$$

Expand \left(5\sqrt{7}\right)^{2}.

$$\frac{7\sqrt{5}-5\sqrt{7}}{245-5^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{7\sqrt{5}-5\sqrt{7}}{245-25\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{7\sqrt{5}-5\sqrt{7}}{245-25\times 7}$$

Multiply 25 and 7 to get 175.

$$\frac{7\sqrt{5}-5\sqrt{7}}{245-175}$$

Subtract 175 from 245 to get 70.

$$\frac{7\sqrt{5}-5\sqrt{7}}{70}$$

Factor

$$\frac{\sqrt{7} {(\sqrt{5} \sqrt{7} - 5)}}{70} = 0.03462456124536539$$