Solve frac{7+sqrt{7}}{{left(sqrt{7}+1right)}^2}

$$\frac{ 7+ \sqrt{ 7 } }{ { \left( \sqrt{ 7 } +1 \right) }^{ 2 } }$$

Evaluate

$$\frac{7-\sqrt{7}}{6}\approx 0.725708115$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{7}+1\right)^{2}.

$$\frac{7+\sqrt{7}}{\left(\sqrt{7}\right)^{2}+2\sqrt{7}+1}$$

The square of \sqrt{7} is 7.

$$\frac{7+\sqrt{7}}{7+2\sqrt{7}+1}$$

Add 7 and 1 to get 8.

$$\frac{7+\sqrt{7}}{8+2\sqrt{7}}$$

Rationalize the denominator of \frac{7+\sqrt{7}}{8+2\sqrt{7}} by multiplying numerator and denominator by 8-2\sqrt{7}.

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

Consider \left(8+2\sqrt{7}\right)\left(8-2\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}.

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

Calculate 8 to the power of 2 and get 64.

$$\frac{\left(7+\sqrt{7}\right)\left(8-2\sqrt{7}\right)}{64-\left(2\sqrt{7}\right)^{2}}$$

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

$$\frac{\left(7+\sqrt{7}\right)\left(8-2\sqrt{7}\right)}{64-2^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(7+\sqrt{7}\right)\left(8-2\sqrt{7}\right)}{64-4\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(7+\sqrt{7}\right)\left(8-2\sqrt{7}\right)}{64-4\times 7}$$

Multiply 4 and 7 to get 28.

$$\frac{\left(7+\sqrt{7}\right)\left(8-2\sqrt{7}\right)}{64-28}$$

Subtract 28 from 64 to get 36.

$$\frac{\left(7+\sqrt{7}\right)\left(8-2\sqrt{7}\right)}{36}$$

Use the distributive property to multiply 7+\sqrt{7} by 8-2\sqrt{7} and combine like terms.

$$\frac{56-6\sqrt{7}-2\left(\sqrt{7}\right)^{2}}{36}$$

The square of \sqrt{7} is 7.

$$\frac{56-6\sqrt{7}-2\times 7}{36}$$

Multiply -2 and 7 to get -14.

$$\frac{56-6\sqrt{7}-14}{36}$$

Subtract 14 from 56 to get 42.

$$\frac{42-6\sqrt{7}}{36}$$

Factor

$$\frac{\sqrt{7} {(\sqrt{7} - 1)}}{6} = 0.7257081148225684$$