Solve frac{sqrt{8}}{sqrt{21}}*frac{sqrt{56}}{sqrt{3}}

$$\frac { \sqrt { 8 } } { \sqrt { 21 } } \times \frac { \sqrt { 56 } } { \sqrt { 3 } }$$

Evaluate

$$\frac{8}{3}\approx 2.666666667$$

Solution Steps

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}.

$$\frac{2\sqrt{2}}{\sqrt{21}}\times \frac{\sqrt{56}}{\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{2\sqrt{2}\sqrt{21}}{\left(\sqrt{21}\right)^{2}}\times \frac{\sqrt{56}}{\sqrt{3}}$$

The square of \sqrt{21} is 21.

$$\frac{2\sqrt{2}\sqrt{21}}{21}\times \frac{\sqrt{56}}{\sqrt{3}}$$

To multiply \sqrt{2} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{2\sqrt{42}}{21}\times \frac{\sqrt{56}}{\sqrt{3}}$$

Factor 56=2^{2}\times 14. Rewrite the square root of the product \sqrt{2^{2}\times 14} as the product of square roots \sqrt{2^{2}}\sqrt{14}. Take the square root of 2^{2}.

$$\frac{2\sqrt{42}}{21}\times \frac{2\sqrt{14}}{\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{14}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{42}}{21}\times \frac{2\sqrt{14}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{42}}{21}\times \frac{2\sqrt{14}\sqrt{3}}{3}$$

To multiply \sqrt{14} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{42}}{21}\times \frac{2\sqrt{42}}{3}$$

Multiply \frac{2\sqrt{42}}{21} times \frac{2\sqrt{42}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{42}\times 2\sqrt{42}}{21\times 3}$$

Multiply \sqrt{42} and \sqrt{42} to get 42.

$$\frac{2\times 42\times 2}{21\times 3}$$

Cancel out 21 in both numerator and denominator.

$$\frac{2\times 2\times 2}{3}$$

Multiply 2 and 2 to get 4.

$$\frac{4\times 2}{3}$$

Multiply 4 and 2 to get 8.

$$\frac{8}{3}$$

Factor

$$\frac{2 ^ {3}}{3} = 2\frac{2}{3} = 2.6666666666666665$$