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

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

Evaluate

$$2\sqrt{5}+3\approx 7.472135955$$

Solution Steps

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

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

Consider \left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

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

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

The square of \sqrt{5} is 5.

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

Add 7 and 5 to get 12.

$$\frac{7\sqrt{5}+12+\sqrt{5}}{4}$$

Combine 7\sqrt{5} and \sqrt{5} to get 8\sqrt{5}.

$$\frac{8\sqrt{5}+12}{4}$$

Divide each term of 8\sqrt{5}+12 by 4 to get 2\sqrt{5}+3.

$$2\sqrt{5}+3$$