Solve 49/3sqrt{24/343}:sqrt{1/42}

$$\frac { 49 } { 3 } \sqrt { \frac { 24 } { 343 } } : \sqrt { \frac { 1 } { 42 } }$$

Evaluate

$$28$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{24}{343}} as the division of square roots \frac{\sqrt{24}}{\sqrt{343}}.

$$\frac{\frac{49}{3}\times \frac{\sqrt{24}}{\sqrt{343}}}{\sqrt{\frac{1}{42}}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{\frac{49}{3}\times \frac{2\sqrt{6}}{\sqrt{343}}}{\sqrt{\frac{1}{42}}}$$

Factor 343=7^{2}\times 7. Rewrite the square root of the product \sqrt{7^{2}\times 7} as the product of square roots \sqrt{7^{2}}\sqrt{7}. Take the square root of 7^{2}.

$$\frac{\frac{49}{3}\times \frac{2\sqrt{6}}{7\sqrt{7}}}{\sqrt{\frac{1}{42}}}$$

Rationalize the denominator of \frac{2\sqrt{6}}{7\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\frac{49}{3}\times \frac{2\sqrt{6}\sqrt{7}}{7\left(\sqrt{7}\right)^{2}}}{\sqrt{\frac{1}{42}}}$$

The square of \sqrt{7} is 7.

$$\frac{\frac{49}{3}\times \frac{2\sqrt{6}\sqrt{7}}{7\times 7}}{\sqrt{\frac{1}{42}}}$$

To multiply \sqrt{6} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\frac{49}{3}\times \frac{2\sqrt{42}}{7\times 7}}{\sqrt{\frac{1}{42}}}$$

Multiply 7 and 7 to get 49.

$$\frac{\frac{49}{3}\times \frac{2\sqrt{42}}{49}}{\sqrt{\frac{1}{42}}}$$

Multiply \frac{49}{3} times \frac{2\sqrt{42}}{49} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{49\times 2\sqrt{42}}{3\times 49}}{\sqrt{\frac{1}{42}}}$$

Cancel out 49 in both numerator and denominator.

$$\frac{\frac{2\sqrt{42}}{3}}{\sqrt{\frac{1}{42}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{42}} as the division of square roots \frac{\sqrt{1}}{\sqrt{42}}.

$$\frac{\frac{2\sqrt{42}}{3}}{\frac{\sqrt{1}}{\sqrt{42}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{2\sqrt{42}}{3}}{\frac{1}{\sqrt{42}}}$$

Rationalize the denominator of \frac{1}{\sqrt{42}} by multiplying numerator and denominator by \sqrt{42}.

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

The square of \sqrt{42} is 42.

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

Divide \frac{2\sqrt{42}}{3} by \frac{\sqrt{42}}{42} by multiplying \frac{2\sqrt{42}}{3} by the reciprocal of \frac{\sqrt{42}}{42}.

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

Cancel out 3\sqrt{42} in both numerator and denominator.

$$2\times 14$$

Multiply 2 and 14 to get 28.

$$28$$

Factor

$$2^{2}\times 7$$