Solve sqrt{50/9}+sqrt{18/16}-5sqrt{200/36}

$$\sqrt { \frac { 50 } { 9 } } + \sqrt { \frac { 18 } { 16 } } - 5 \sqrt { \frac { 200 } { 36 } }$$

Evaluate

$$-\frac{71\sqrt{2}}{12}\approx -8.367430244$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{50}{9}} as the division of square roots \frac{\sqrt{50}}{\sqrt{9}}.

$$\frac{\sqrt{50}}{\sqrt{9}}+\sqrt{\frac{18}{16}}-5\sqrt{\frac{200}{36}}$$

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}.

$$\frac{5\sqrt{2}}{\sqrt{9}}+\sqrt{\frac{18}{16}}-5\sqrt{\frac{200}{36}}$$

Calculate the square root of 9 and get 3.

$$\frac{5\sqrt{2}}{3}+\sqrt{\frac{18}{16}}-5\sqrt{\frac{200}{36}}$$

Reduce the fraction \frac{18}{16} to lowest terms by extracting and canceling out 2.

$$\frac{5\sqrt{2}}{3}+\sqrt{\frac{9}{8}}-5\sqrt{\frac{200}{36}}$$

Rewrite the square root of the division \sqrt{\frac{9}{8}} as the division of square roots \frac{\sqrt{9}}{\sqrt{8}}.

$$\frac{5\sqrt{2}}{3}+\frac{\sqrt{9}}{\sqrt{8}}-5\sqrt{\frac{200}{36}}$$

Calculate the square root of 9 and get 3.

$$\frac{5\sqrt{2}}{3}+\frac{3}{\sqrt{8}}-5\sqrt{\frac{200}{36}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{5\sqrt{2}}{3}+\frac{3}{2\sqrt{2}}-5\sqrt{\frac{200}{36}}$$

Rationalize the denominator of \frac{3}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{5\sqrt{2}}{3}+\frac{3\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-5\sqrt{\frac{200}{36}}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}}{3}+\frac{3\sqrt{2}}{2\times 2}-5\sqrt{\frac{200}{36}}$$

Multiply 2 and 2 to get 4.

$$\frac{5\sqrt{2}}{3}+\frac{3\sqrt{2}}{4}-5\sqrt{\frac{200}{36}}$$

Combine \frac{5\sqrt{2}}{3} and \frac{3\sqrt{2}}{4} to get \frac{29}{12}\sqrt{2}.

$$\frac{29}{12}\sqrt{2}-5\sqrt{\frac{200}{36}}$$

Reduce the fraction \frac{200}{36} to lowest terms by extracting and canceling out 4.

$$\frac{29}{12}\sqrt{2}-5\sqrt{\frac{50}{9}}$$

Rewrite the square root of the division \sqrt{\frac{50}{9}} as the division of square roots \frac{\sqrt{50}}{\sqrt{9}}.

$$\frac{29}{12}\sqrt{2}-5\times \frac{\sqrt{50}}{\sqrt{9}}$$

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}.

$$\frac{29}{12}\sqrt{2}-5\times \frac{5\sqrt{2}}{\sqrt{9}}$$

Calculate the square root of 9 and get 3.

$$\frac{29}{12}\sqrt{2}-5\times \frac{5\sqrt{2}}{3}$$

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

$$\frac{29}{12}\sqrt{2}-\frac{5\times 5\sqrt{2}}{3}$$

Multiply 5 and 5 to get 25.

$$\frac{29}{12}\sqrt{2}-\frac{25\sqrt{2}}{3}$$

Combine \frac{29}{12}\sqrt{2} and -\frac{25\sqrt{2}}{3} to get -\frac{71}{12}\sqrt{2}.

$$-\frac{71}{12}\sqrt{2}$$