Solve -sqrt{5/3}-sqrt{5/54}

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

Evaluate

$$-\frac{\sqrt{15}}{3}-\frac{\sqrt{30}}{18}\approx -1.595284758$$

Short Solution Steps

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, subtract them by subtracting their numerators.

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