Solve 5sqrt{(243/32)}

$$5 \sqrt{ ( \frac{ 243 }{ 32 } ) }$$

Evaluate

$$\frac{45\sqrt{6}}{8}\approx 13.778379803$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{243}{32}} as the division of square roots \frac{\sqrt{243}}{\sqrt{32}}.

$$5\times \frac{\sqrt{243}}{\sqrt{32}}$$

Factor 243=9^{2}\times 3. Rewrite the square root of the product \sqrt{9^{2}\times 3} as the product of square roots \sqrt{9^{2}}\sqrt{3}. Take the square root of 9^{2}.

$$5\times \frac{9\sqrt{3}}{\sqrt{32}}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$5\times \frac{9\sqrt{3}}{4\sqrt{2}}$$

Rationalize the denominator of \frac{9\sqrt{3}}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$5\times \frac{9\sqrt{3}\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$5\times \frac{9\sqrt{3}\sqrt{2}}{4\times 2}$$

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

$$5\times \frac{9\sqrt{6}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$5\times \frac{9\sqrt{6}}{8}$$

Express 5\times \frac{9\sqrt{6}}{8} as a single fraction.

$$\frac{5\times 9\sqrt{6}}{8}$$

Multiply 5 and 9 to get 45.

$$\frac{45\sqrt{6}}{8}$$