Solve frac{3sqrt{5}}{1+2sqrt{5}}

$$\frac { 3 \sqrt { 5 } } { 1 + 2 \sqrt { 5 } }$$

Evaluate

$$\frac{30-3\sqrt{5}}{19}\approx 1.225884004$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{5}}{1+2\sqrt{5}} by multiplying numerator and denominator by 1-2\sqrt{5}.

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

Consider \left(1+2\sqrt{5}\right)\left(1-2\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}.

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

Calculate 1 to the power of 2 and get 1.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-20}$$

Subtract 20 from 1 to get -19.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{-19}$$

Use the distributive property to multiply 3\sqrt{5} by 1-2\sqrt{5}.

$$\frac{3\sqrt{5}-6\left(\sqrt{5}\right)^{2}}{-19}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}-6\times 5}{-19}$$

Multiply -6 and 5 to get -30.

$$\frac{3\sqrt{5}-30}{-19}$$