Solve sqrt{49*(100-64)*169/sqrt{81*196*sqrt{625}}:frac{sqrt{144}+25}{12}}=

$$\sqrt { \frac { 49 \cdot ( 100 - 64 ) \cdot 169 } { \sqrt { 81 } \cdot 196 \cdot \sqrt { 625 } } : \frac { \sqrt { 144 } + 25 } { 12 } } =$$

Evaluate

$$\frac{26\sqrt{111}}{185}\approx 1.480686473$$

Solution Steps

Divide \frac{49\left(100-64\right)\times 169}{196\sqrt{81}\sqrt{625}} by \frac{\sqrt{144}+25}{12} by multiplying \frac{49\left(100-64\right)\times 169}{196\sqrt{81}\sqrt{625}} by the reciprocal of \frac{\sqrt{144}+25}{12}.

$$\sqrt{\frac{49\left(100-64\right)\times 169\times 12}{196\sqrt{81}\sqrt{625}\left(\sqrt{144}+25\right)}}$$

Cancel out 4\times 49 in both numerator and denominator.

$$\sqrt{\frac{3\times 169\left(100-64\right)}{\sqrt{81}\sqrt{625}\left(\sqrt{144}+25\right)}}$$

Multiply 3 and 169 to get 507.

$$\sqrt{\frac{507\left(100-64\right)}{\sqrt{81}\sqrt{625}\left(\sqrt{144}+25\right)}}$$

Subtract 64 from 100 to get 36.

$$\sqrt{\frac{507\times 36}{\sqrt{81}\sqrt{625}\left(\sqrt{144}+25\right)}}$$

Multiply 507 and 36 to get 18252.

$$\sqrt{\frac{18252}{\sqrt{81}\sqrt{625}\left(\sqrt{144}+25\right)}}$$

Calculate the square root of 81 and get 9.

$$\sqrt{\frac{18252}{9\sqrt{625}\left(\sqrt{144}+25\right)}}$$

Calculate the square root of 625 and get 25.

$$\sqrt{\frac{18252}{9\times 25\left(\sqrt{144}+25\right)}}$$

Multiply 9 and 25 to get 225.

$$\sqrt{\frac{18252}{225\left(\sqrt{144}+25\right)}}$$

Calculate the square root of 144 and get 12.

$$\sqrt{\frac{18252}{225\left(12+25\right)}}$$

Add 12 and 25 to get 37.

$$\sqrt{\frac{18252}{225\times 37}}$$

Multiply 225 and 37 to get 8325.

$$\sqrt{\frac{18252}{8325}}$$

Reduce the fraction \frac{18252}{8325} to lowest terms by extracting and canceling out 9.

$$\sqrt{\frac{2028}{925}}$$

Rewrite the square root of the division \sqrt{\frac{2028}{925}} as the division of square roots \frac{\sqrt{2028}}{\sqrt{925}}.

$$\frac{\sqrt{2028}}{\sqrt{925}}$$

Factor 2028=26^{2}\times 3. Rewrite the square root of the product \sqrt{26^{2}\times 3} as the product of square roots \sqrt{26^{2}}\sqrt{3}. Take the square root of 26^{2}.

$$\frac{26\sqrt{3}}{\sqrt{925}}$$

Factor 925=5^{2}\times 37. Rewrite the square root of the product \sqrt{5^{2}\times 37} as the product of square roots \sqrt{5^{2}}\sqrt{37}. Take the square root of 5^{2}.

$$\frac{26\sqrt{3}}{5\sqrt{37}}$$

Rationalize the denominator of \frac{26\sqrt{3}}{5\sqrt{37}} by multiplying numerator and denominator by \sqrt{37}.

$$\frac{26\sqrt{3}\sqrt{37}}{5\left(\sqrt{37}\right)^{2}}$$

The square of \sqrt{37} is 37.

$$\frac{26\sqrt{3}\sqrt{37}}{5\times 37}$$

To multiply \sqrt{3} and \sqrt{37}, multiply the numbers under the square root.

$$\frac{26\sqrt{111}}{5\times 37}$$

Multiply 5 and 37 to get 185.

$$\frac{26\sqrt{111}}{185}$$