Solve 2/3sqrt{16/25}+2/5sqrt{4/5}-frac{2}{7}sqrt{frac{49}{4}}-frac{3}{5}sqrt{frac{25}{9}}

$$\frac { 2 } { 3 } \sqrt { \frac { 16 } { 25 } } + \frac { 2 } { 5 } \sqrt { \frac { 4 } { 5 } } - \frac { 2 } { 7 } \sqrt { \frac { 49 } { 4 } } - \frac { 3 } { 5 } \sqrt { \frac { 25 } { 9 } }$$

Evaluate

$$\frac{4\sqrt{5}}{25}-\frac{22}{15}\approx -1.10889579$$

Short Solution Steps

Rewrite the square root of the division \frac{16}{25} as the division of square roots \frac{\sqrt{16}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$\frac{2}{3}\times \frac{4}{5}+\frac{2}{5}\sqrt{\frac{4}{5}}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Multiply \frac{2}{3} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 4}{3\times 5}+\frac{2}{5}\sqrt{\frac{4}{5}}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Do the multiplications in the fraction \frac{2\times 4}{3\times 5}.

$$\frac{8}{15}+\frac{2}{5}\sqrt{\frac{4}{5}}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Rewrite the square root of the division \sqrt{\frac{4}{5}} as the division of square roots \frac{\sqrt{4}}{\sqrt{5}}.

$$\frac{8}{15}+\frac{2}{5}\times \frac{\sqrt{4}}{\sqrt{5}}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Calculate the square root of 4 and get 2.

$$\frac{8}{15}+\frac{2}{5}\times \frac{2}{\sqrt{5}}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

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

$$\frac{8}{15}+\frac{2}{5}\times \frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

The square of \sqrt{5} is 5.

$$\frac{8}{15}+\frac{2}{5}\times \frac{2\sqrt{5}}{5}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Multiply \frac{2}{5} times \frac{2\sqrt{5}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}-\frac{2}{7}\sqrt{\frac{49}{4}}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Rewrite the square root of the division \frac{49}{4} as the division of square roots \frac{\sqrt{49}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{8}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}-\frac{2}{7}\times \frac{7}{2}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Multiply -\frac{2}{7} times \frac{7}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}+\frac{-2\times 7}{7\times 2}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Cancel out 7 in both numerator and denominator.

$$\frac{8}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}+\frac{-2}{2}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Divide -2 by 2 to get -1.

$$\frac{8}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}-1-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Convert 1 to fraction \frac{15}{15}.

$$\frac{8}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}-\frac{15}{15}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Since \frac{8}{15} and \frac{15}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8-15}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Subtract 15 from 8 to get -7.

$$-\frac{7}{15}+\frac{2\times 2\sqrt{5}}{5\times 5}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Multiply 2 and 2 to get 4.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{5\times 5}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Multiply 5 and 5 to get 25.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{25}-\frac{3}{5}\sqrt{\frac{25}{9}}$$

Rewrite the square root of the division \frac{25}{9} as the division of square roots \frac{\sqrt{25}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{25}-\frac{3}{5}\times \frac{5}{3}$$

Multiply -\frac{3}{5} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{25}+\frac{-3\times 5}{5\times 3}$$

Cancel out 5 in both numerator and denominator.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{25}+\frac{-3}{3}$$

Divide -3 by 3 to get -1.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{25}-1$$

Convert 1 to fraction \frac{15}{15}.

$$-\frac{7}{15}+\frac{4\sqrt{5}}{25}-\frac{15}{15}$$

Since -\frac{7}{15} and \frac{15}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-7-15}{15}+\frac{4\sqrt{5}}{25}$$

Subtract 15 from -7 to get -22.

$$-\frac{22}{15}+\frac{4\sqrt{5}}{25}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15 and 25 is 75. Multiply -\frac{22}{15} times \frac{5}{5}. Multiply \frac{4\sqrt{5}}{25} times \frac{3}{3}.

$$-\frac{22\times 5}{75}+\frac{3\times 4\sqrt{5}}{75}$$

Since -\frac{22\times 5}{75} and \frac{3\times 4\sqrt{5}}{75} have the same denominator, add them by adding their numerators.

$$\frac{-22\times 5+3\times 4\sqrt{5}}{75}$$

Do the multiplications in -22\times 5+3\times 4\sqrt{5}.

$$\frac{-110+12\sqrt{5}}{75}$$

Factor

$$\frac{2 {(6 \sqrt{5} - 55)}}{75} = -1.1088957902667003$$