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

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

Evaluate

$$\frac{8\sqrt{7}}{7}+2\approx 5.023715784$$

Solution Steps

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

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

The square of \sqrt{7} is 7.

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

Add 7 and 1 to get 8.

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

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

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

The square of \sqrt{7} is 7.

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

Use the distributive property to multiply 8+2\sqrt{7} by \sqrt{7}.

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

The square of \sqrt{7} is 7.

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

Multiply 2 and 7 to get 14.

$$\frac{8\sqrt{7}+14}{7}$$