Solve sqrt{frac{3*8.31*323/6.0210^230.6*8}}{sqrt{3*8.31*323/6.0210^230.6*7}}

$$\frac{ \sqrt{ \frac{ 3 \cdot 8.31 \cdot 323 }{ 6.02 { 10 }^{ 23 } 0.6 \cdot 8 } } }{ \sqrt{ \frac{ 3 \cdot 8.31 \cdot 323 }{ 6.02 { 10 }^{ 23 } 0.6 \cdot 7 } } }$$

Evaluate

$$\frac{\sqrt{14}}{4}\approx 0.935414347$$

Solution Steps

Multiply 3 and 8.31 to get 24.93.

$$\frac{\sqrt{\frac{24.93\times 323}{6.02\times 10^{23}\times 0.6\times 8}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 24.93 and 323 to get 8052.39.

$$\frac{\sqrt{\frac{8052.39}{6.02\times 10^{23}\times 0.6\times 8}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Calculate 10 to the power of 23 and get 100000000000000000000000.

$$\frac{\sqrt{\frac{8052.39}{6.02\times 100000000000000000000000\times 0.6\times 8}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 6.02 and 100000000000000000000000 to get 602000000000000000000000.

$$\frac{\sqrt{\frac{8052.39}{602000000000000000000000\times 0.6\times 8}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 602000000000000000000000 and 0.6 to get 361200000000000000000000.

$$\frac{\sqrt{\frac{8052.39}{361200000000000000000000\times 8}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 361200000000000000000000 and 8 to get 2889600000000000000000000.

$$\frac{\sqrt{\frac{8052.39}{2889600000000000000000000}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Expand \frac{8052.39}{2889600000000000000000000} by multiplying both numerator and the denominator by 100.

$$\frac{\sqrt{\frac{805239}{288960000000000000000000000}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Reduce the fraction \frac{805239}{288960000000000000000000000} to lowest terms by extracting and canceling out 3.

$$\frac{\sqrt{\frac{268413}{96320000000000000000000000}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Rewrite the square root of the division \sqrt{\frac{268413}{96320000000000000000000000}} as the division of square roots \frac{\sqrt{268413}}{\sqrt{96320000000000000000000000}}.

$$\frac{\frac{\sqrt{268413}}{\sqrt{96320000000000000000000000}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Factor 96320000000000000000000000=400000000000^{2}\times 602. Rewrite the square root of the product \sqrt{400000000000^{2}\times 602} as the product of square roots \sqrt{400000000000^{2}}\sqrt{602}. Take the square root of 400000000000^{2}.

$$\frac{\frac{\sqrt{268413}}{400000000000\sqrt{602}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Rationalize the denominator of \frac{\sqrt{268413}}{400000000000\sqrt{602}} by multiplying numerator and denominator by \sqrt{602}.

$$\frac{\frac{\sqrt{268413}\sqrt{602}}{400000000000\left(\sqrt{602}\right)^{2}}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

The square of \sqrt{602} is 602.

$$\frac{\frac{\sqrt{268413}\sqrt{602}}{400000000000\times 602}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

To multiply \sqrt{268413} and \sqrt{602}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{161584626}}{400000000000\times 602}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 400000000000 and 602 to get 240800000000000.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{3\times 8.31\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 3 and 8.31 to get 24.93.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{24.93\times 323}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Multiply 24.93 and 323 to get 8052.39.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{8052.39}{6.02\times 10^{23}\times 0.6\times 7}}}$$

Calculate 10 to the power of 23 and get 100000000000000000000000.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{8052.39}{6.02\times 100000000000000000000000\times 0.6\times 7}}}$$

Multiply 6.02 and 100000000000000000000000 to get 602000000000000000000000.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{8052.39}{602000000000000000000000\times 0.6\times 7}}}$$

Multiply 602000000000000000000000 and 0.6 to get 361200000000000000000000.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{8052.39}{361200000000000000000000\times 7}}}$$

Multiply 361200000000000000000000 and 7 to get 2528400000000000000000000.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{8052.39}{2528400000000000000000000}}}$$

Expand \frac{8052.39}{2528400000000000000000000} by multiplying both numerator and the denominator by 100.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{805239}{252840000000000000000000000}}}$$

Reduce the fraction \frac{805239}{252840000000000000000000000} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\sqrt{\frac{268413}{84280000000000000000000000}}}$$

Rewrite the square root of the division \sqrt{\frac{268413}{84280000000000000000000000}} as the division of square roots \frac{\sqrt{268413}}{\sqrt{84280000000000000000000000}}.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\frac{\sqrt{268413}}{\sqrt{84280000000000000000000000}}}$$

Factor 84280000000000000000000000=1400000000000^{2}\times 43. Rewrite the square root of the product \sqrt{1400000000000^{2}\times 43} as the product of square roots \sqrt{1400000000000^{2}}\sqrt{43}. Take the square root of 1400000000000^{2}.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\frac{\sqrt{268413}}{1400000000000\sqrt{43}}}$$

Rationalize the denominator of \frac{\sqrt{268413}}{1400000000000\sqrt{43}} by multiplying numerator and denominator by \sqrt{43}.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\frac{\sqrt{268413}\sqrt{43}}{1400000000000\left(\sqrt{43}\right)^{2}}}$$

The square of \sqrt{43} is 43.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\frac{\sqrt{268413}\sqrt{43}}{1400000000000\times 43}}$$

To multiply \sqrt{268413} and \sqrt{43}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\frac{\sqrt{11541759}}{1400000000000\times 43}}$$

Multiply 1400000000000 and 43 to get 60200000000000.

$$\frac{\frac{\sqrt{161584626}}{240800000000000}}{\frac{\sqrt{11541759}}{60200000000000}}$$

Divide \frac{\sqrt{161584626}}{240800000000000} by \frac{\sqrt{11541759}}{60200000000000} by multiplying \frac{\sqrt{161584626}}{240800000000000} by the reciprocal of \frac{\sqrt{11541759}}{60200000000000}.

$$\frac{\sqrt{161584626}\times 60200000000000}{240800000000000\sqrt{11541759}}$$

Cancel out 60200000000000 in both numerator and denominator.

$$\frac{\sqrt{161584626}}{4\sqrt{11541759}}$$

Rationalize the denominator of \frac{\sqrt{161584626}}{4\sqrt{11541759}} by multiplying numerator and denominator by \sqrt{11541759}.

$$\frac{\sqrt{161584626}\sqrt{11541759}}{4\left(\sqrt{11541759}\right)^{2}}$$

The square of \sqrt{11541759} is 11541759.

$$\frac{\sqrt{161584626}\sqrt{11541759}}{4\times 11541759}$$

Factor 161584626=11541759\times 14. Rewrite the square root of the product \sqrt{11541759\times 14} as the product of square roots \sqrt{11541759}\sqrt{14}.

$$\frac{\sqrt{11541759}\sqrt{14}\sqrt{11541759}}{4\times 11541759}$$

Multiply \sqrt{11541759} and \sqrt{11541759} to get 11541759.

$$\frac{11541759\sqrt{14}}{4\times 11541759}$$

Multiply 4 and 11541759 to get 46167036.

$$\frac{11541759\sqrt{14}}{46167036}$$

Divide 11541759\sqrt{14} by 46167036 to get \frac{1}{4}\sqrt{14}.

$$\frac{1}{4}\sqrt{14}$$