Solve -5sqrt{45}*3/2sqrt{2/3}

$$- 5 \sqrt { 45 } \times \frac { 3 } { 2 } \sqrt { \frac { 2 } { 3 } }$$

Evaluate

$$-\frac{15\sqrt{30}}{2}\approx -41.079191813$$

Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

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

Multiply -5 and 3 to get -15.

$$-15\sqrt{5}\times \frac{3}{2}\sqrt{\frac{2}{3}}$$

Express -15\times \frac{3}{2} as a single fraction.

$$\frac{-15\times 3}{2}\sqrt{5}\sqrt{\frac{2}{3}}$$

Multiply -15 and 3 to get -45.

$$\frac{-45}{2}\sqrt{5}\sqrt{\frac{2}{3}}$$

Fraction \frac{-45}{2} can be rewritten as -\frac{45}{2} by extracting the negative sign.

$$-\frac{45}{2}\sqrt{5}\sqrt{\frac{2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$-\frac{45}{2}\sqrt{5}\times \frac{\sqrt{2}}{\sqrt{3}}$$

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

$$-\frac{45}{2}\sqrt{5}\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$-\frac{45}{2}\sqrt{5}\times \frac{\sqrt{2}\sqrt{3}}{3}$$

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

$$-\frac{45}{2}\sqrt{5}\times \frac{\sqrt{6}}{3}$$

Multiply -\frac{45}{2} times \frac{\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-45\sqrt{6}}{2\times 3}\sqrt{5}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-15\sqrt{6}}{2}\sqrt{5}$$

Express \frac{-15\sqrt{6}}{2}\sqrt{5} as a single fraction.

$$\frac{-15\sqrt{6}\sqrt{5}}{2}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{-15\sqrt{30}}{2}$$