Solve 3sqrt{2}{left(1/2sqrt{10}right)}^div2divsqrt{15}

$$3 \sqrt{ 2 } { \left( \frac{ 1 }{ 2 } \sqrt{ 10 } \right) }^{ } \div 2 \div \sqrt{ 15 }$$

Evaluate

$$\frac{\sqrt{3}}{2}\approx 0.866025404$$

Solution Steps

Express \frac{\frac{3\sqrt{2}\times \left(\frac{1}{2}\sqrt{10}\right)^{1}}{2}}{\sqrt{15}} as a single fraction.

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

Calculate \frac{1}{2}\sqrt{10} to the power of 1 and get \frac{1}{2}\sqrt{10}.

$$\frac{3\sqrt{2}\times \frac{1}{2}\sqrt{10}}{2\sqrt{15}}$$

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

$$\frac{\frac{3}{2}\sqrt{2}\sqrt{10}}{2\sqrt{15}}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Cancel out 2 in both numerator and denominator.

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

Rationalize the denominator of \frac{\frac{3}{2}\sqrt{5}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\frac{3}{2}\sqrt{5}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

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

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply \frac{3}{2} and 5 to get \frac{15}{2}.

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

Divide \frac{15}{2}\sqrt{3} by 15 to get \frac{1}{2}\sqrt{3}.

$$\frac{1}{2}\sqrt{3}$$