Solve frac{2sqrt{5}}{2sqrt{5}+4}frac{2sqrt{5}-4}{2sqrt{5}-4}

$$\frac{ 2 \sqrt{ 5 } }{ 2 \sqrt{ 5 } +4 } \frac{ 2 \sqrt{ 5 } -4 }{ 2 \sqrt{ 5 } -4 }$$

Evaluate

$$5-2\sqrt{5}\approx 0.527864045$$

Solution Steps

Divide 2\sqrt{5}-4 by 2\sqrt{5}-4 to get 1.

$$\frac{2\sqrt{5}}{2\sqrt{5}+4}\times 1$$

Rationalize the denominator of \frac{2\sqrt{5}}{2\sqrt{5}+4} by multiplying numerator and denominator by 2\sqrt{5}-4.

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

Consider \left(2\sqrt{5}+4\right)\left(2\sqrt{5}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\frac{2\sqrt{5}\left(2\sqrt{5}-4\right)}{20-4^{2}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{2\sqrt{5}\left(2\sqrt{5}-4\right)}{20-16}\times 1$$

Subtract 16 from 20 to get 4.

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

Divide 2\sqrt{5}\left(2\sqrt{5}-4\right) by 4 to get \frac{1}{2}\sqrt{5}\left(2\sqrt{5}-4\right).

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

Use the distributive property to multiply \frac{1}{2}\sqrt{5} by 2\sqrt{5}-4.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

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

Cancel out 2 and 2.

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

Multiply \frac{1}{2} and -4 to get \frac{-4}{2}.

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

Divide -4 by 2 to get -2.

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

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

$$5-2\sqrt{5}$$

Factor

$$\sqrt{5} {(\sqrt{5} - 2)} = 0.527864045$$