Solve (3/2sqrt{5}-sqrt{2}quad)div(1/2sqrt{3})

$$( \frac { 3 } { 2 } \sqrt { 5 } - \sqrt { 2 } \quad ) \div ( \frac { 1 } { 2 } \sqrt { 3 } )$$

Evaluate

$$-\frac{2\sqrt{6}}{3}+\sqrt{15}\approx 2.239990184$$

Solution Steps

Rationalize the denominator of \frac{\frac{3}{2}\sqrt{5}-\sqrt{2}}{\frac{1}{2}\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

Divide \left(\frac{3}{2}\sqrt{5}-\sqrt{2}\right)\sqrt{3} by \frac{3}{2} by multiplying \left(\frac{3}{2}\sqrt{5}-\sqrt{2}\right)\sqrt{3} by the reciprocal of \frac{3}{2}.

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

Use the distributive property to multiply \frac{3}{2}\sqrt{5}-\sqrt{2} by \sqrt{3}.

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

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

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\left(\frac{3}{2}\sqrt{15}-\sqrt{6}\right)\times 2}{3}$$

Use the distributive property to multiply \frac{3}{2}\sqrt{15}-\sqrt{6} by 2.

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

Cancel out 2 and 2.

$$\frac{3\sqrt{15}-2\sqrt{6}}{3}$$

Factor

$$\frac{\sqrt{3} {(3 \sqrt{5} - 2 \sqrt{2})}}{3} = 2.239990184351965$$