Solve sqrt{3/4}div(-1/2sqrt{12/3})

$$\sqrt { \frac { 3 } { 4 } } \div ( - \frac { 1 } { 2 } \sqrt { 1 \frac { 2 } { 3 } } )$$

Evaluate

$$-\frac{3\sqrt{5}}{5}\approx -1.341640786$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$\frac{\frac{\sqrt{3}}{\sqrt{4}}}{-\frac{1}{2}\sqrt{\frac{3+2}{3}}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}\sqrt{\frac{3+2}{3}}}$$

Add 3 and 2 to get 5.

$$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}\sqrt{\frac{5}{3}}}$$

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

$$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}\times \frac{\sqrt{5}}{\sqrt{3}}}$$

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

$$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}\times \frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}\times \frac{\sqrt{5}\sqrt{3}}{3}}$$

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

$$\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}\times \frac{\sqrt{15}}{3}}$$

Multiply -\frac{1}{2} times \frac{\sqrt{15}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\sqrt{3}}{2}}{\frac{-\sqrt{15}}{2\times 3}}$$

Divide \frac{\sqrt{3}}{2} by \frac{-\sqrt{15}}{2\times 3} by multiplying \frac{\sqrt{3}}{2} by the reciprocal of \frac{-\sqrt{15}}{2\times 3}.

$$\frac{\sqrt{3}\times 2\times 3}{2\left(-1\right)\sqrt{15}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{3\sqrt{3}}{-\sqrt{15}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{-3\sqrt{3}}{\sqrt{15}}$$

Rationalize the denominator of \frac{-3\sqrt{3}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

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

The square of \sqrt{15} is 15.

$$\frac{-3\sqrt{3}\sqrt{15}}{15}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Divide -3\times 3\sqrt{5} by 15 to get -\frac{1}{5}\times 3\sqrt{5}.

$$-\frac{1}{5}\times 3\sqrt{5}$$

Express -\frac{1}{5}\times 3 as a single fraction.

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

Fraction \frac{-3}{5} can be rewritten as -\frac{3}{5} by extracting the negative sign.

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