Solve (-sqrt{12/3})+sqrt{5/54}

$$( - \sqrt { 1 \frac { 2 } { 3 } } ) + \sqrt { \frac { 5 } { 54 } }$$

Evaluate

$$\frac{\sqrt{30}}{18}-\frac{\sqrt{15}}{3}\approx -0.986704139$$

Short Solution Steps

Multiply 1 and 3 to get 3.

$$-\sqrt{\frac{3+2}{3}}+\sqrt{\frac{5}{54}}$$

Add 3 and 2 to get 5.

$$-\sqrt{\frac{5}{3}}+\sqrt{\frac{5}{54}}$$

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

$$-\frac{\sqrt{5}}{\sqrt{3}}+\sqrt{\frac{5}{54}}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$-\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{\frac{5}{54}}$$

The square of \sqrt{3} is 3.

$$-\frac{\sqrt{5}\sqrt{3}}{3}+\sqrt{\frac{5}{54}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$-\frac{\sqrt{15}}{3}+\sqrt{\frac{5}{54}}$$

Rewrite the square root of the division \sqrt{\frac{5}{54}} as the division of square roots \frac{\sqrt{5}}{\sqrt{54}}.

$$-\frac{\sqrt{15}}{3}+\frac{\sqrt{5}}{\sqrt{54}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$-\frac{\sqrt{15}}{3}+\frac{\sqrt{5}}{3\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{5}}{3\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$-\frac{\sqrt{15}}{3}+\frac{\sqrt{5}\sqrt{6}}{3\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$-\frac{\sqrt{15}}{3}+\frac{\sqrt{5}\sqrt{6}}{3\times 6}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$-\frac{\sqrt{15}}{3}+\frac{\sqrt{30}}{3\times 6}$$

Multiply 3 and 6 to get 18.

$$-\frac{\sqrt{15}}{3}+\frac{\sqrt{30}}{18}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 18 is 18. Multiply -\frac{\sqrt{15}}{3} times \frac{6}{6}.

$$-\frac{6\sqrt{15}}{18}+\frac{\sqrt{30}}{18}$$

Since -\frac{6\sqrt{15}}{18} and \frac{\sqrt{30}}{18} have the same denominator, add them by adding their numerators.

$$\frac{-6\sqrt{15}+\sqrt{30}}{18}$$