Solve 2/3sqrt{20}*(1/3sqrt{48})divsqrt{22/3}

$$\frac { 2 } { 3 } \sqrt { 20 } \times ( \frac { 1 } { 3 } \sqrt { 48 } ) \div \sqrt { 2 \frac { 2 } { 3 } }$$

Evaluate

$$\frac{4\sqrt{10}}{3}\approx 4.216370214$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\frac{2}{3}\times 2\sqrt{5}\times \frac{1}{3}\sqrt{48}}{\sqrt{\frac{2\times 3+2}{3}}}$$

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

$$\frac{\frac{2\times 2}{3}\sqrt{5}\times \frac{1}{3}\sqrt{48}}{\sqrt{\frac{2\times 3+2}{3}}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{4}{3}\sqrt{5}\times \frac{1}{3}\sqrt{48}}{\sqrt{\frac{2\times 3+2}{3}}}$$

Multiply \frac{4}{3} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{4\times 1}{3\times 3}\sqrt{5}\sqrt{48}}{\sqrt{\frac{2\times 3+2}{3}}}$$

Do the multiplications in the fraction \frac{4\times 1}{3\times 3}.

$$\frac{\frac{4}{9}\sqrt{5}\sqrt{48}}{\sqrt{\frac{2\times 3+2}{3}}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

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

Express \frac{4}{9}\times 4 as a single fraction.

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

Multiply 4 and 4 to get 16.

$$\frac{\frac{16}{9}\sqrt{5}\sqrt{3}}{\sqrt{\frac{2\times 3+2}{3}}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{16}{9}\sqrt{15}}{\sqrt{\frac{2\times 3+2}{3}}}$$

Multiply 2 and 3 to get 6.

$$\frac{\frac{16}{9}\sqrt{15}}{\sqrt{\frac{6+2}{3}}}$$

Add 6 and 2 to get 8.

$$\frac{\frac{16}{9}\sqrt{15}}{\sqrt{\frac{8}{3}}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\frac{16}{9}\sqrt{15}}{\frac{\sqrt{8}}{\sqrt{3}}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\frac{16}{9}\sqrt{15}}{\frac{2\sqrt{2}}{\sqrt{3}}}$$

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

$$\frac{\frac{16}{9}\sqrt{15}}{\frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{16}{9}\sqrt{15}}{\frac{2\sqrt{2}\sqrt{3}}{3}}$$

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

$$\frac{\frac{16}{9}\sqrt{15}}{\frac{2\sqrt{6}}{3}}$$

Divide \frac{16}{9}\sqrt{15} by \frac{2\sqrt{6}}{3} by multiplying \frac{16}{9}\sqrt{15} by the reciprocal of \frac{2\sqrt{6}}{3}.

$$\frac{\frac{16}{9}\sqrt{15}\times 3}{2\sqrt{6}}$$

Rationalize the denominator of \frac{\frac{16}{9}\sqrt{15}\times 3}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\frac{16}{9}\sqrt{15}\times 3\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\frac{16}{9}\sqrt{15}\times 3\sqrt{6}}{2\times 6}$$

Express \frac{16}{9}\times 3 as a single fraction.

$$\frac{\frac{16\times 3}{9}\sqrt{15}\sqrt{6}}{2\times 6}$$

Multiply 16 and 3 to get 48.

$$\frac{\frac{48}{9}\sqrt{15}\sqrt{6}}{2\times 6}$$

Reduce the fraction \frac{48}{9} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{16}{3}\sqrt{15}\sqrt{6}}{2\times 6}$$

To multiply \sqrt{15} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\frac{16}{3}\sqrt{90}}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$\frac{\frac{16}{3}\sqrt{90}}{12}$$

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

$$\frac{\frac{16}{3}\times 3\sqrt{10}}{12}$$

Cancel out 3 and 3.

$$\frac{16\sqrt{10}}{12}$$

Divide 16\sqrt{10} by 12 to get \frac{4}{3}\sqrt{10}.

$$\frac{4}{3}\sqrt{10}$$