Solve 2/5sqrt{20}-3/6sqrt{80}+1/8sqrt{180}+6sqrt{4}

$$\frac { 2 } { 5 } \sqrt { 20 } - \frac { 3 } { 6 } \sqrt { 80 } + \frac { 1 } { 8 } \sqrt { 180 } + 6 \sqrt { 4 }$$

Evaluate

$$-\frac{9\sqrt{5}}{20}+12\approx 10.99376941$$

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}.

$$\frac{2}{5}\times 2\sqrt{5}-\frac{3}{6}\sqrt{80}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

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

$$\frac{2\times 2}{5}\sqrt{5}-\frac{3}{6}\sqrt{80}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

Multiply 2 and 2 to get 4.

$$\frac{4}{5}\sqrt{5}-\frac{3}{6}\sqrt{80}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{4}{5}\sqrt{5}-\frac{1}{2}\sqrt{80}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{4}{5}\sqrt{5}-\frac{1}{2}\times 4\sqrt{5}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

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

$$\frac{4}{5}\sqrt{5}-\frac{4}{2}\sqrt{5}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

Divide 4 by 2 to get 2.

$$\frac{4}{5}\sqrt{5}-2\sqrt{5}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

Combine \frac{4}{5}\sqrt{5} and -2\sqrt{5} to get -\frac{6}{5}\sqrt{5}.

$$-\frac{6}{5}\sqrt{5}+\frac{1}{8}\sqrt{180}+6\sqrt{4}$$

Factor 180=6^{2}\times 5. Rewrite the square root of the product \sqrt{6^{2}\times 5} as the product of square roots \sqrt{6^{2}}\sqrt{5}. Take the square root of 6^{2}.

$$-\frac{6}{5}\sqrt{5}+\frac{1}{8}\times 6\sqrt{5}+6\sqrt{4}$$

Multiply \frac{1}{8} and 6 to get \frac{6}{8}.

$$-\frac{6}{5}\sqrt{5}+\frac{6}{8}\sqrt{5}+6\sqrt{4}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$-\frac{6}{5}\sqrt{5}+\frac{3}{4}\sqrt{5}+6\sqrt{4}$$

Combine -\frac{6}{5}\sqrt{5} and \frac{3}{4}\sqrt{5} to get -\frac{9}{20}\sqrt{5}.

$$-\frac{9}{20}\sqrt{5}+6\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$-\frac{9}{20}\sqrt{5}+6\times 2$$

Multiply 6 and 2 to get 12.

$$-\frac{9}{20}\sqrt{5}+12$$

Factor

$$\frac{3 {(80 - 3 \sqrt{5})}}{20} = 10.993769410125093$$