Solve (sqrt{20}+sqrt{5})div(sqrt{5}-2)

$$( \sqrt { 20 } + \sqrt { 5 } ) \div ( \sqrt { 5 } - 2 )$$

Evaluate

$$6\sqrt{5}+15\approx 28.416407865$$

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}.

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

Combine 2\sqrt{5} and \sqrt{5} to get 3\sqrt{5}.

$$\frac{3\sqrt{5}}{\sqrt{5}-2}$$

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

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

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

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

Square \sqrt{5}. Square 2.

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

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

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

$$3\left(\sqrt{5}\right)^{2}+6\sqrt{5}$$

The square of \sqrt{5} is 5.

$$3\times 5+6\sqrt{5}$$

Multiply 3 and 5 to get 15.

$$15+6\sqrt{5}$$