Solve 9sqrt{5/48}div(-3/2sqrt{41/12})

$$9 \sqrt { \frac { 5 } { 48 } } \div ( - \frac { 3 } { 2 } \sqrt { 4 \frac { 1 } { 12 } } )$$

Evaluate

$$-\frac{3\sqrt{5}}{7}\approx -0.958314847$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{48}} as the division of square roots \frac{\sqrt{5}}{\sqrt{48}}.

$$\frac{9\times \frac{\sqrt{5}}{\sqrt{48}}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{9\times \frac{\sqrt{5}}{4\sqrt{3}}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

Rationalize the denominator of \frac{\sqrt{5}}{4\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{9\times \frac{\sqrt{5}\sqrt{3}}{4\left(\sqrt{3}\right)^{2}}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

The square of \sqrt{3} is 3.

$$\frac{9\times \frac{\sqrt{5}\sqrt{3}}{4\times 3}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

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

$$\frac{9\times \frac{\sqrt{15}}{4\times 3}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

Multiply 4 and 3 to get 12.

$$\frac{9\times \frac{\sqrt{15}}{12}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

Express 9\times \frac{\sqrt{15}}{12} as a single fraction.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\sqrt{\frac{4\times 12+1}{12}}}$$

Multiply 4 and 12 to get 48.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\sqrt{\frac{48+1}{12}}}$$

Add 48 and 1 to get 49.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\sqrt{\frac{49}{12}}}$$

Rewrite the square root of the division \sqrt{\frac{49}{12}} as the division of square roots \frac{\sqrt{49}}{\sqrt{12}}.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\times \frac{\sqrt{49}}{\sqrt{12}}}$$

Calculate the square root of 49 and get 7.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\times \frac{7}{\sqrt{12}}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\times \frac{7}{2\sqrt{3}}}$$

Rationalize the denominator of \frac{7}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\times \frac{7\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\times \frac{7\sqrt{3}}{2\times 3}}$$

Multiply 2 and 3 to get 6.

$$\frac{\frac{9\sqrt{15}}{12}}{-\frac{3}{2}\times \frac{7\sqrt{3}}{6}}$$

Multiply -\frac{3}{2} times \frac{7\sqrt{3}}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{9\sqrt{15}}{12}}{\frac{-3\times 7\sqrt{3}}{2\times 6}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{9\sqrt{15}}{12}}{\frac{-7\sqrt{3}}{2\times 2}}$$

Divide \frac{9\sqrt{15}}{12} by \frac{-7\sqrt{3}}{2\times 2} by multiplying \frac{9\sqrt{15}}{12} by the reciprocal of \frac{-7\sqrt{3}}{2\times 2}.

$$\frac{9\sqrt{15}\times 2\times 2}{12\left(-1\right)\times 7\sqrt{3}}$$

Cancel out 2\times 2\times 3 in both numerator and denominator.

$$\frac{3\sqrt{15}}{-7\sqrt{3}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{-3\sqrt{15}}{7\sqrt{3}}$$

Rationalize the denominator of \frac{-3\sqrt{15}}{7\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{-3\sqrt{15}\sqrt{3}}{7\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{-3\sqrt{15}\sqrt{3}}{7\times 3}$$

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}.

$$\frac{-3\sqrt{3}\sqrt{5}\sqrt{3}}{7\times 3}$$

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

$$\frac{-3\times 3\sqrt{5}}{7\times 3}$$

Multiply 7 and 3 to get 21.

$$\frac{-3\times 3\sqrt{5}}{21}$$

Divide -3\times 3\sqrt{5} by 21 to get -\frac{1}{7}\times 3\sqrt{5}.

$$-\frac{1}{7}\times 3\sqrt{5}$$

Express -\frac{1}{7}\times 3 as a single fraction.

$$\frac{-3}{7}\sqrt{5}$$

Fraction \frac{-3}{7} can be rewritten as -\frac{3}{7} by extracting the negative sign.

$$-\frac{3}{7}\sqrt{5}$$