Solve (1/2sqrt{5}+2)(2/5sqrt{5}+4)

$$( \frac { 1 } { 2 } \sqrt { 5 } + 2 ) ( \frac { 2 } { 5 } \sqrt { 5 } + 4 )$$

Evaluate

$$\frac{14\sqrt{5}}{5}+9\approx 15.260990337$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{2}\sqrt{5}+2 by each term of \frac{2}{5}\sqrt{5}+4.

$$\frac{1}{2}\sqrt{5}\times \frac{2}{5}\sqrt{5}+\frac{1}{2}\sqrt{5}\times 4+2\times \frac{2}{5}\sqrt{5}+8$$

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

$$\frac{1}{2}\times 5\times \frac{2}{5}+\frac{1}{2}\sqrt{5}\times 4+2\times \frac{2}{5}\sqrt{5}+8$$

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

$$\frac{5}{2}\times \frac{2}{5}+\frac{1}{2}\sqrt{5}\times 4+2\times \frac{2}{5}\sqrt{5}+8$$

Cancel out \frac{5}{2} and its reciprocal \frac{2}{5}.

$$1+\frac{1}{2}\sqrt{5}\times 4+2\times \frac{2}{5}\sqrt{5}+8$$

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

$$1+\frac{4}{2}\sqrt{5}+2\times \frac{2}{5}\sqrt{5}+8$$

Divide 4 by 2 to get 2.

$$1+2\sqrt{5}+2\times \frac{2}{5}\sqrt{5}+8$$

Express 2\times \frac{2}{5} as a single fraction.

$$1+2\sqrt{5}+\frac{2\times 2}{5}\sqrt{5}+8$$

Multiply 2 and 2 to get 4.

$$1+2\sqrt{5}+\frac{4}{5}\sqrt{5}+8$$

Combine 2\sqrt{5} and \frac{4}{5}\sqrt{5} to get \frac{14}{5}\sqrt{5}.

$$1+\frac{14}{5}\sqrt{5}+8$$

Add 1 and 8 to get 9.

$$9+\frac{14}{5}\sqrt{5}$$

Factor

$$\frac{14 \sqrt{5} + 45}{5} = 15.260990336999413$$