Solve 1/1+3sqrt{5}+1/1-3sqrt{5}

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

Evaluate

$$-\frac{1}{22}\approx -0.045454545$$

Short Solution Steps

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

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

Consider \left(1+3\sqrt{5}\right)\left(1-3\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}.

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

Calculate 1 to the power of 2 and get 1.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{5} is 5.

$$\frac{1-3\sqrt{5}}{1-9\times 5}+\frac{1}{1-3\sqrt{5}}$$

Multiply 9 and 5 to get 45.

$$\frac{1-3\sqrt{5}}{1-45}+\frac{1}{1-3\sqrt{5}}$$

Subtract 45 from 1 to get -44.

$$\frac{1-3\sqrt{5}}{-44}+\frac{1}{1-3\sqrt{5}}$$

Multiply both numerator and denominator by -1.

$$\frac{-1+3\sqrt{5}}{44}+\frac{1}{1-3\sqrt{5}}$$

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

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{\left(1-3\sqrt{5}\right)\left(1+3\sqrt{5}\right)}$$

Consider \left(1-3\sqrt{5}\right)\left(1+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}.

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{1^{2}-\left(-3\sqrt{5}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{1-\left(-3\sqrt{5}\right)^{2}}$$

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

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{1-\left(-3\right)^{2}\left(\sqrt{5}\right)^{2}}$$

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

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{1-9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{1-9\times 5}$$

Multiply 9 and 5 to get 45.

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{1-45}$$

Subtract 45 from 1 to get -44.

$$\frac{-1+3\sqrt{5}}{44}+\frac{1+3\sqrt{5}}{-44}$$

Multiply both numerator and denominator by -1.

$$\frac{-1+3\sqrt{5}}{44}+\frac{-1-3\sqrt{5}}{44}$$

Since \frac{-1+3\sqrt{5}}{44} and \frac{-1-3\sqrt{5}}{44} have the same denominator, add them by adding their numerators.

$$\frac{-1+3\sqrt{5}-1-3\sqrt{5}}{44}$$

Do the calculations in -1+3\sqrt{5}-1-3\sqrt{5}.

$$\frac{-2}{44}$$

Reduce the fraction \frac{-2}{44} to lowest terms by extracting and canceling out 2.

$$-\frac{1}{22}$$

Factor

$$-\frac{1}{22} = -0.045454545454545456$$