Solve frac{2sqrt{4}sqrt{20}}{sqrt{5}}+(frac{0,5sqrt{5}+sqrt{5}}{6}

$$\frac { 2 \sqrt { 4 } \sqrt { 20 } } { \sqrt { 5 } } + ( \frac { 0,5 \sqrt { 5 } + \sqrt { 5 } } { 6 }$$

Evaluate

$$\frac{\sqrt{5}}{4}+8\approx 8.559016994$$

Solution Steps

Factor 20=4\times 5. Rewrite the square root of the product \sqrt{4\times 5} as the product of square roots \sqrt{4}\sqrt{5}.

$$\frac{2\sqrt{4}\sqrt{4}\sqrt{5}}{\sqrt{5}}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

Multiply \sqrt{4} and \sqrt{4} to get 4.

$$\frac{2\times 4\sqrt{5}}{\sqrt{5}}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

Multiply 2 and 4 to get 8.

$$\frac{8\sqrt{5}}{\sqrt{5}}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

Rationalize the denominator of \frac{8\sqrt{5}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{8\sqrt{5}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

The square of \sqrt{5} is 5.

$$\frac{8\sqrt{5}\sqrt{5}}{5}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

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

$$\frac{8\times 5}{5}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

Multiply 8 and 5 to get 40.

$$\frac{40}{5}+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

Divide 40 by 5 to get 8.

$$8+\frac{0,5\sqrt{5}+\sqrt{5}}{6}$$

Combine 0,5\sqrt{5} and \sqrt{5} to get 1,5\sqrt{5}.

$$8+\frac{1,5\sqrt{5}}{6}$$

Divide 1,5\sqrt{5} by 6 to get 0,25\sqrt{5}.

$$8+0,25\sqrt{5}$$

Factor

$$\frac{\sqrt{5} + 32}{4} = 8.559016994374947$$