Solve 12sqrt{1/6}div3sqrt{7/12}*1/2sqrt{101/2}

$$12 \sqrt { \frac { 1 } { 6 } } \div 3 \sqrt { \frac { 7 } { 12 } } \times \frac { 1 } { 2 } \sqrt { 10 \frac { 1 } { 2 } }$$

Evaluate

$$\frac{7\sqrt{3}}{6}\approx 2.020725942$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{6}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6}}.

$$\frac{12\times \frac{\sqrt{1}}{\sqrt{6}}}{3}\sqrt{\frac{7}{12}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Calculate the square root of 1 and get 1.

$$\frac{12\times \frac{1}{\sqrt{6}}}{3}\sqrt{\frac{7}{12}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{12\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}}{3}\sqrt{\frac{7}{12}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

The square of \sqrt{6} is 6.

$$\frac{12\times \frac{\sqrt{6}}{6}}{3}\sqrt{\frac{7}{12}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Cancel out 6, the greatest common factor in 12 and 6.

$$\frac{2\sqrt{6}}{3}\sqrt{\frac{7}{12}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{7}{12}} as the division of square roots \frac{\sqrt{7}}{\sqrt{12}}.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{7}}{\sqrt{12}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{7}}{2\sqrt{3}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Rationalize the denominator of \frac{\sqrt{7}}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{7}\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{7}\sqrt{3}}{2\times 3}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{2\times 3}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Multiply 2 and 3 to get 6.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\sqrt{\frac{10\times 2+1}{2}}$$

Multiply 10 and 2 to get 20.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\sqrt{\frac{20+1}{2}}$$

Add 20 and 1 to get 21.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\sqrt{\frac{21}{2}}$$

Rewrite the square root of the division \sqrt{\frac{21}{2}} as the division of square roots \frac{\sqrt{21}}{\sqrt{2}}.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\times \frac{\sqrt{21}}{\sqrt{2}}$$

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

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\times \frac{\sqrt{21}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\times \frac{\sqrt{21}\sqrt{2}}{2}$$

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

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{21}}{6}\times \frac{1}{2}\times \frac{\sqrt{42}}{2}$$

Multiply \frac{2\sqrt{6}}{3} times \frac{\sqrt{21}}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{6}\sqrt{21}}{3\times 6}\times \frac{1}{2}\times \frac{\sqrt{42}}{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{6}\sqrt{21}}{3\times 3}\times \frac{1}{2}\times \frac{\sqrt{42}}{2}$$

Multiply \frac{\sqrt{6}\sqrt{21}}{3\times 3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}\sqrt{21}}{3\times 3\times 2}\times \frac{\sqrt{42}}{2}$$

Multiply \frac{\sqrt{6}\sqrt{21}}{3\times 3\times 2} times \frac{\sqrt{42}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}\sqrt{21}\sqrt{42}}{3\times 3\times 2\times 2}$$

Factor 42=6\times 7. Rewrite the square root of the product \sqrt{6\times 7} as the product of square roots \sqrt{6}\sqrt{7}.

$$\frac{\sqrt{6}\sqrt{21}\sqrt{6}\sqrt{7}}{3\times 3\times 2\times 2}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{6\sqrt{21}\sqrt{7}}{3\times 3\times 2\times 2}$$

Factor 21=7\times 3. Rewrite the square root of the product \sqrt{7\times 3} as the product of square roots \sqrt{7}\sqrt{3}.

$$\frac{6\sqrt{7}\sqrt{3}\sqrt{7}}{3\times 3\times 2\times 2}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$\frac{6\times 7\sqrt{3}}{3\times 3\times 2\times 2}$$

Multiply 6 and 7 to get 42.

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

Multiply 3 and 3 to get 9.

$$\frac{42\sqrt{3}}{9\times 2\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{42\sqrt{3}}{18\times 2}$$

Multiply 18 and 2 to get 36.

$$\frac{42\sqrt{3}}{36}$$

Divide 42\sqrt{3} by 36 to get \frac{7}{6}\sqrt{3}.

$$\frac{7}{6}\sqrt{3}$$