Solve frac{sqrt{5}+1}{7-3sqrt{5}}

$$\frac { \sqrt { 5 } + 1 } { 7 - 3 \sqrt { 5 } }$$

Evaluate

$$\frac{5\sqrt{5}+11}{2}\approx 11.090169944$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}+1}{7-3\sqrt{5}} by multiplying numerator and denominator by 7+3\sqrt{5}.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{\left(7-3\sqrt{5}\right)\left(7+3\sqrt{5}\right)}$$

Consider \left(7-3\sqrt{5}\right)\left(7+3\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}.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{7^{2}-\left(-3\sqrt{5}\right)^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{49-\left(-3\sqrt{5}\right)^{2}}$$

Expand \left(-3\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{49-\left(-3\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{49-9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{49-9\times 5}$$

Multiply 9 and 5 to get 45.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{49-45}$$

Subtract 45 from 49 to get 4.

$$\frac{\left(\sqrt{5}+1\right)\left(7+3\sqrt{5}\right)}{4}$$

Apply the distributive property by multiplying each term of \sqrt{5}+1 by each term of 7+3\sqrt{5}.

$$\frac{7\sqrt{5}+3\left(\sqrt{5}\right)^{2}+7+3\sqrt{5}}{4}$$

The square of \sqrt{5} is 5.

$$\frac{7\sqrt{5}+3\times 5+7+3\sqrt{5}}{4}$$

Multiply 3 and 5 to get 15.

$$\frac{7\sqrt{5}+15+7+3\sqrt{5}}{4}$$

Add 15 and 7 to get 22.

$$\frac{7\sqrt{5}+22+3\sqrt{5}}{4}$$

Combine 7\sqrt{5} and 3\sqrt{5} to get 10\sqrt{5}.

$$\frac{10\sqrt{5}+22}{4}$$