Solve frac{2sqrt{3}}{3sqrt{2}}*5^3/6sqrt{3}

$$\frac { 2 \sqrt { 3 } } { 3 \sqrt { 2 } } \times \frac { 5 ^ { 3 } } { 6 \sqrt { 3 } }$$

Evaluate

$$\frac{125\sqrt{2}}{18}\approx 9.820927516$$

Solution Steps

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

$$\frac{2\sqrt{3}\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}\times \frac{5^{3}}{6\sqrt{3}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{3}\sqrt{2}}{3\times 2}\times \frac{5^{3}}{6\sqrt{3}}$$

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

$$\frac{2\sqrt{6}}{3\times 2}\times \frac{5^{3}}{6\sqrt{3}}$$

Multiply 3 and 2 to get 6.

$$\frac{2\sqrt{6}}{6}\times \frac{5^{3}}{6\sqrt{3}}$$

Divide 2\sqrt{6} by 6 to get \frac{1}{3}\sqrt{6}.

$$\frac{1}{3}\sqrt{6}\times \frac{5^{3}}{6\sqrt{3}}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{1}{3}\sqrt{6}\times \frac{125}{6\sqrt{3}}$$

Rationalize the denominator of \frac{125}{6\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{1}{3}\sqrt{6}\times \frac{125\sqrt{3}}{6\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{1}{3}\sqrt{6}\times \frac{125\sqrt{3}}{6\times 3}$$

Multiply 6 and 3 to get 18.

$$\frac{1}{3}\sqrt{6}\times \frac{125\sqrt{3}}{18}$$

Multiply \frac{1}{3} times \frac{125\sqrt{3}}{18} by multiplying numerator times numerator and denominator times denominator.

$$\frac{125\sqrt{3}}{3\times 18}\sqrt{6}$$

Multiply 3 and 18 to get 54.

$$\frac{125\sqrt{3}}{54}\sqrt{6}$$

Express \frac{125\sqrt{3}}{54}\sqrt{6} as a single fraction.

$$\frac{125\sqrt{3}\sqrt{6}}{54}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{125\sqrt{3}\sqrt{3}\sqrt{2}}{54}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{125\times 3\sqrt{2}}{54}$$

Multiply 125 and 3 to get 375.

$$\frac{375\sqrt{2}}{54}$$

Divide 375\sqrt{2} by 54 to get \frac{125}{18}\sqrt{2}.

$$\frac{125}{18}\sqrt{2}$$