Solve 3sqrt{2/9}-5sqrt{2/9}-5sqrt{50}+2/3sqrt{2/25}

$$3 \sqrt { \frac { 2 } { 9 } } - 5 \sqrt { \frac { 2 } { 9 } } - 5 \sqrt { 50 } + \frac { 2 } { 3 } \sqrt { \frac { 2 } { 25 } }$$

Evaluate

$$-\frac{383\sqrt{2}}{15}\approx -36.109586293$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{9}} as the division of square roots \frac{\sqrt{2}}{\sqrt{9}}.

$$3\times \frac{\sqrt{2}}{\sqrt{9}}-5\sqrt{\frac{2}{9}}-5\sqrt{50}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Calculate the square root of 9 and get 3.

$$3\times \frac{\sqrt{2}}{3}-5\sqrt{\frac{2}{9}}-5\sqrt{50}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Cancel out 3 and 3.

$$\sqrt{2}-5\sqrt{\frac{2}{9}}-5\sqrt{50}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Rewrite the square root of the division \sqrt{\frac{2}{9}} as the division of square roots \frac{\sqrt{2}}{\sqrt{9}}.

$$\sqrt{2}-5\times \frac{\sqrt{2}}{\sqrt{9}}-5\sqrt{50}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Calculate the square root of 9 and get 3.

$$\sqrt{2}-5\times \frac{\sqrt{2}}{3}-5\sqrt{50}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Express -5\times \frac{\sqrt{2}}{3} as a single fraction.

$$\sqrt{2}+\frac{-5\sqrt{2}}{3}-5\sqrt{50}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

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

Multiply -5 and 5 to get -25.

$$\sqrt{2}+\frac{-5\sqrt{2}}{3}-25\sqrt{2}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Combine \sqrt{2} and -25\sqrt{2} to get -24\sqrt{2}.

$$-24\sqrt{2}+\frac{-5\sqrt{2}}{3}+\frac{2}{3}\sqrt{\frac{2}{25}}$$

Rewrite the square root of the division \sqrt{\frac{2}{25}} as the division of square roots \frac{\sqrt{2}}{\sqrt{25}}.

$$-24\sqrt{2}+\frac{-5\sqrt{2}}{3}+\frac{2}{3}\times \frac{\sqrt{2}}{\sqrt{25}}$$

Calculate the square root of 25 and get 5.

$$-24\sqrt{2}+\frac{-5\sqrt{2}}{3}+\frac{2}{3}\times \frac{\sqrt{2}}{5}$$

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

$$-24\sqrt{2}+\frac{-5\sqrt{2}}{3}+\frac{2\sqrt{2}}{3\times 5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -24\sqrt{2} times \frac{3}{3}.

$$\frac{3\left(-24\right)\sqrt{2}}{3}+\frac{-5\sqrt{2}}{3}+\frac{2\sqrt{2}}{3\times 5}$$

Since \frac{3\left(-24\right)\sqrt{2}}{3} and \frac{-5\sqrt{2}}{3} have the same denominator, add them by adding their numerators.

$$\frac{3\left(-24\right)\sqrt{2}-5\sqrt{2}}{3}+\frac{2\sqrt{2}}{3\times 5}$$

Do the multiplications in 3\left(-24\right)\sqrt{2}-5\sqrt{2}.

$$\frac{-72\sqrt{2}-5\sqrt{2}}{3}+\frac{2\sqrt{2}}{3\times 5}$$

Do the calculations in -72\sqrt{2}-5\sqrt{2}.

$$\frac{-77\sqrt{2}}{3}+\frac{2\sqrt{2}}{3\times 5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 3\times 5 is 3\times 5. Multiply \frac{-77\sqrt{2}}{3} times \frac{5}{5}.

$$\frac{5\left(-77\right)\sqrt{2}}{3\times 5}+\frac{2\sqrt{2}}{3\times 5}$$

Since \frac{5\left(-77\right)\sqrt{2}}{3\times 5} and \frac{2\sqrt{2}}{3\times 5} have the same denominator, add them by adding their numerators.

$$\frac{5\left(-77\right)\sqrt{2}+2\sqrt{2}}{3\times 5}$$

Do the multiplications in 5\left(-77\right)\sqrt{2}+2\sqrt{2}.

$$\frac{-385\sqrt{2}+2\sqrt{2}}{3\times 5}$$

Do the calculations in -385\sqrt{2}+2\sqrt{2}.

$$\frac{-383\sqrt{2}}{3\times 5}$$

Expand 3\times 5.

$$\frac{-383\sqrt{2}}{15}$$