Solve (frac{3sqrt{15}}{4}-sqrt{12})divfrac{sqrt{3}}{2}

$$( \frac { 3 \sqrt { 15 } } { 4 } - \sqrt { 12 } ) \div \frac { \sqrt { 3 } } { 2 }$$

Evaluate

$$\frac{3\sqrt{5}}{2}-4\approx -0.645898034$$

Short Solution Steps

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}.

$$\frac{\frac{3\sqrt{15}}{4}-2\sqrt{3}}{\frac{\sqrt{3}}{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{3} times \frac{4}{4}.

$$\frac{\frac{3\sqrt{15}}{4}+\frac{4\left(-2\right)\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}}$$

Since \frac{3\sqrt{15}}{4} and \frac{4\left(-2\right)\sqrt{3}}{4} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\sqrt{15}+4\left(-2\right)\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}}$$

Do the multiplications in 3\sqrt{15}+4\left(-2\right)\sqrt{3}.

$$\frac{\frac{3\sqrt{15}-8\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}}$$

Divide \frac{3\sqrt{15}-8\sqrt{3}}{4} by \frac{\sqrt{3}}{2} by multiplying \frac{3\sqrt{15}-8\sqrt{3}}{4} by the reciprocal of \frac{\sqrt{3}}{2}.

$$\frac{\left(3\sqrt{15}-8\sqrt{3}\right)\times 2}{4\sqrt{3}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-8\sqrt{3}+3\sqrt{15}}{2\sqrt{3}}$$

Rationalize the denominator of \frac{-8\sqrt{3}+3\sqrt{15}}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(-8\sqrt{3}+3\sqrt{15}\right)\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(-8\sqrt{3}+3\sqrt{15}\right)\sqrt{3}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{\left(-8\sqrt{3}+3\sqrt{15}\right)\sqrt{3}}{6}$$

Use the distributive property to multiply -8\sqrt{3}+3\sqrt{15} by \sqrt{3}.

$$\frac{-8\left(\sqrt{3}\right)^{2}+3\sqrt{15}\sqrt{3}}{6}$$

The square of \sqrt{3} is 3.

$$\frac{-8\times 3+3\sqrt{15}\sqrt{3}}{6}$$

Multiply -8 and 3 to get -24.

$$\frac{-24+3\sqrt{15}\sqrt{3}}{6}$$

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}.

$$\frac{-24+3\sqrt{3}\sqrt{5}\sqrt{3}}{6}$$

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

$$\frac{-24+3\times 3\sqrt{5}}{6}$$

Multiply 3 and 3 to get 9.

$$\frac{-24+9\sqrt{5}}{6}$$

Factor

$$\frac{3 \sqrt{5} - 8}{2} = -0.6458980337503153$$