Solve 30/5-sqrt{20}

$$\frac { 30 } { 5 - \sqrt { 20 } }$$

Evaluate

$$12\sqrt{5}+30\approx 56.83281573$$

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}.

$$\frac{30}{5-2\sqrt{5}}$$

Rationalize the denominator of \frac{30}{5-2\sqrt{5}} by multiplying numerator and denominator by 5+2\sqrt{5}.

$$\frac{30\left(5+2\sqrt{5}\right)}{\left(5-2\sqrt{5}\right)\left(5+2\sqrt{5}\right)}$$

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

$$\frac{30\left(5+2\sqrt{5}\right)}{5^{2}-\left(-2\sqrt{5}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{30\left(5+2\sqrt{5}\right)}{25-\left(-2\sqrt{5}\right)^{2}}$$

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

$$\frac{30\left(5+2\sqrt{5}\right)}{25-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{30\left(5+2\sqrt{5}\right)}{25-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{30\left(5+2\sqrt{5}\right)}{25-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{30\left(5+2\sqrt{5}\right)}{25-20}$$

Subtract 20 from 25 to get 5.

$$\frac{30\left(5+2\sqrt{5}\right)}{5}$$

Divide 30\left(5+2\sqrt{5}\right) by 5 to get 6\left(5+2\sqrt{5}\right).

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

Use the distributive property to multiply 6 by 5+2\sqrt{5}.

$$30+12\sqrt{5}$$