Solve frac{6-sqrt{20}}{6+sqrt{20}}

$$\frac { 6 - \sqrt { 20 } } { 6 + \sqrt { 20 } }$$

Evaluate

$$\frac{7-3\sqrt{5}}{2}\approx 0.145898034$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{6-2\sqrt{5}}{6+\sqrt{20}}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{6-2\sqrt{5}}{6+2\sqrt{5}}$$

Rationalize the denominator of \frac{6-2\sqrt{5}}{6+2\sqrt{5}} by multiplying numerator and denominator by 6-2\sqrt{5}.

$$\frac{\left(6-2\sqrt{5}\right)\left(6-2\sqrt{5}\right)}{\left(6+2\sqrt{5}\right)\left(6-2\sqrt{5}\right)}$$

Consider \left(6+2\sqrt{5}\right)\left(6-2\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}.

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

Multiply 6-2\sqrt{5} and 6-2\sqrt{5} to get \left(6-2\sqrt{5}\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-2\sqrt{5}\right)^{2}.

$$\frac{36-24\sqrt{5}+4\left(\sqrt{5}\right)^{2}}{6^{2}-\left(2\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{36-24\sqrt{5}+4\times 5}{6^{2}-\left(2\sqrt{5}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{36-24\sqrt{5}+20}{6^{2}-\left(2\sqrt{5}\right)^{2}}$$

Add 36 and 20 to get 56.

$$\frac{56-24\sqrt{5}}{6^{2}-\left(2\sqrt{5}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{56-24\sqrt{5}}{36-\left(2\sqrt{5}\right)^{2}}$$

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

$$\frac{56-24\sqrt{5}}{36-2^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{56-24\sqrt{5}}{36-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{56-24\sqrt{5}}{36-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{56-24\sqrt{5}}{36-20}$$

Subtract 20 from 36 to get 16.

$$\frac{56-24\sqrt{5}}{16}$$