Solve 1/2sqrt{10}*(3sqrt{15}-3sqrt{6})

$$\frac { 1 } { 2 } \sqrt { 10 } \times ( 3 \sqrt { 15 } - 3 \sqrt { 6 } )$$

Evaluate

$$\frac{15\sqrt{6}-6\sqrt{15}}{2}\approx 6.752223032$$

Solution Steps

Use the distributive property to multiply \frac{1}{2}\sqrt{10} by 3\sqrt{15}-3\sqrt{6}.

$$\frac{1}{2}\sqrt{10}\times 3\sqrt{15}+\frac{1}{2}\sqrt{10}\left(-3\right)\sqrt{6}$$

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

$$\frac{3}{2}\sqrt{10}\sqrt{15}+\frac{1}{2}\sqrt{10}\left(-3\right)\sqrt{6}$$

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{3}{2}\sqrt{150}+\frac{1}{2}\sqrt{10}\left(-3\right)\sqrt{6}$$

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

$$\frac{3}{2}\sqrt{150}+\frac{-3}{2}\sqrt{10}\sqrt{6}$$

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

$$\frac{3}{2}\sqrt{150}-\frac{3}{2}\sqrt{10}\sqrt{6}$$

To multiply \sqrt{10} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{3}{2}\sqrt{150}-\frac{3}{2}\sqrt{60}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{3}{2}\times 5\sqrt{6}-\frac{3}{2}\sqrt{60}$$

Express \frac{3}{2}\times 5 as a single fraction.

$$\frac{3\times 5}{2}\sqrt{6}-\frac{3}{2}\sqrt{60}$$

Multiply 3 and 5 to get 15.

$$\frac{15}{2}\sqrt{6}-\frac{3}{2}\sqrt{60}$$

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

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

Cancel out 2 and 2.

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

Factor

$$\frac{3 \sqrt{10} {(\sqrt{15} - \sqrt{6})}}{2} = 6.752223032251587$$