Solve 603-38*36/10/sqrt{554-frac{(38)^2{10}}sqrt{810-(36)^2/10}}

$$\frac { 603 - \frac { 38 \times 36 } { 10 } } { \sqrt { 554 - \frac { ( 38 ) ^ { 2 } } { 10 } } \sqrt { 810 - \frac { ( 36 ) ^ { 2 } } { 10 } } }$$

Evaluate

$$\frac{37\sqrt{21}}{192}\approx 0.883100525$$

Solution Steps

Multiply 38 and 36 to get 1368.

$$\frac{603-\frac{1368}{10}}{\sqrt{554-\frac{38^{2}}{10}}\sqrt{810-\frac{36^{2}}{10}}}$$

Reduce the fraction \frac{1368}{10} to lowest terms by extracting and canceling out 2.

$$\frac{603-\frac{684}{5}}{\sqrt{554-\frac{38^{2}}{10}}\sqrt{810-\frac{36^{2}}{10}}}$$

Convert 603 to fraction \frac{3015}{5}.

$$\frac{\frac{3015}{5}-\frac{684}{5}}{\sqrt{554-\frac{38^{2}}{10}}\sqrt{810-\frac{36^{2}}{10}}}$$

Since \frac{3015}{5} and \frac{684}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3015-684}{5}}{\sqrt{554-\frac{38^{2}}{10}}\sqrt{810-\frac{36^{2}}{10}}}$$

Subtract 684 from 3015 to get 2331.

$$\frac{\frac{2331}{5}}{\sqrt{554-\frac{38^{2}}{10}}\sqrt{810-\frac{36^{2}}{10}}}$$

Calculate 38 to the power of 2 and get 1444.

$$\frac{\frac{2331}{5}}{\sqrt{554-\frac{1444}{10}}\sqrt{810-\frac{36^{2}}{10}}}$$

Reduce the fraction \frac{1444}{10} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{2331}{5}}{\sqrt{554-\frac{722}{5}}\sqrt{810-\frac{36^{2}}{10}}}$$

Convert 554 to fraction \frac{2770}{5}.

$$\frac{\frac{2331}{5}}{\sqrt{\frac{2770}{5}-\frac{722}{5}}\sqrt{810-\frac{36^{2}}{10}}}$$

Since \frac{2770}{5} and \frac{722}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2331}{5}}{\sqrt{\frac{2770-722}{5}}\sqrt{810-\frac{36^{2}}{10}}}$$

Subtract 722 from 2770 to get 2048.

$$\frac{\frac{2331}{5}}{\sqrt{\frac{2048}{5}}\sqrt{810-\frac{36^{2}}{10}}}$$

Rewrite the square root of the division \sqrt{\frac{2048}{5}} as the division of square roots \frac{\sqrt{2048}}{\sqrt{5}}.

$$\frac{\frac{2331}{5}}{\frac{\sqrt{2048}}{\sqrt{5}}\sqrt{810-\frac{36^{2}}{10}}}$$

Factor 2048=32^{2}\times 2. Rewrite the square root of the product \sqrt{32^{2}\times 2} as the product of square roots \sqrt{32^{2}}\sqrt{2}. Take the square root of 32^{2}.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{2}}{\sqrt{5}}\sqrt{810-\frac{36^{2}}{10}}}$$

Rationalize the denominator of \frac{32\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\sqrt{810-\frac{36^{2}}{10}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{2}\sqrt{5}}{5}\sqrt{810-\frac{36^{2}}{10}}}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\sqrt{810-\frac{36^{2}}{10}}}$$

Calculate 36 to the power of 2 and get 1296.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\sqrt{810-\frac{1296}{10}}}$$

Reduce the fraction \frac{1296}{10} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\sqrt{810-\frac{648}{5}}}$$

Convert 810 to fraction \frac{4050}{5}.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\sqrt{\frac{4050}{5}-\frac{648}{5}}}$$

Since \frac{4050}{5} and \frac{648}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\sqrt{\frac{4050-648}{5}}}$$

Subtract 648 from 4050 to get 3402.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\sqrt{\frac{3402}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{3402}{5}} as the division of square roots \frac{\sqrt{3402}}{\sqrt{5}}.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\times \frac{\sqrt{3402}}{\sqrt{5}}}$$

Factor 3402=9^{2}\times 42. Rewrite the square root of the product \sqrt{9^{2}\times 42} as the product of square roots \sqrt{9^{2}}\sqrt{42}. Take the square root of 9^{2}.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\times \frac{9\sqrt{42}}{\sqrt{5}}}$$

Rationalize the denominator of \frac{9\sqrt{42}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\times \frac{9\sqrt{42}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\times \frac{9\sqrt{42}\sqrt{5}}{5}}$$

To multiply \sqrt{42} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}}{5}\times \frac{9\sqrt{210}}{5}}$$

Multiply \frac{32\sqrt{10}}{5} times \frac{9\sqrt{210}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2331}{5}}{\frac{32\sqrt{10}\times 9\sqrt{210}}{5\times 5}}$$

Divide \frac{2331}{5} by \frac{32\sqrt{10}\times 9\sqrt{210}}{5\times 5} by multiplying \frac{2331}{5} by the reciprocal of \frac{32\sqrt{10}\times 9\sqrt{210}}{5\times 5}.

$$\frac{2331\times 5\times 5}{5\times 32\sqrt{10}\times 9\sqrt{210}}$$

Cancel out 5\times 9 in both numerator and denominator.

$$\frac{5\times 259}{32\sqrt{10}\sqrt{210}}$$

Rationalize the denominator of \frac{5\times 259}{32\sqrt{10}\sqrt{210}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{5\times 259\sqrt{10}}{32\left(\sqrt{10}\right)^{2}\sqrt{210}}$$

The square of \sqrt{10} is 10.

$$\frac{5\times 259\sqrt{10}}{32\times 10\sqrt{210}}$$

Multiply 5 and 259 to get 1295.

$$\frac{1295\sqrt{10}}{32\times 10\sqrt{210}}$$

Multiply 32 and 10 to get 320.

$$\frac{1295\sqrt{10}}{320\sqrt{210}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{259\sqrt{10}}{64\sqrt{210}}$$

Rationalize the denominator of \frac{259\sqrt{10}}{64\sqrt{210}} by multiplying numerator and denominator by \sqrt{210}.

$$\frac{259\sqrt{10}\sqrt{210}}{64\left(\sqrt{210}\right)^{2}}$$

The square of \sqrt{210} is 210.

$$\frac{259\sqrt{10}\sqrt{210}}{64\times 210}$$

Factor 210=10\times 21. Rewrite the square root of the product \sqrt{10\times 21} as the product of square roots \sqrt{10}\sqrt{21}.

$$\frac{259\sqrt{10}\sqrt{10}\sqrt{21}}{64\times 210}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{259\times 10\sqrt{21}}{64\times 210}$$

Multiply 64 and 210 to get 13440.

$$\frac{259\times 10\sqrt{21}}{13440}$$

Multiply 259 and 10 to get 2590.

$$\frac{2590\sqrt{21}}{13440}$$

Divide 2590\sqrt{21} by 13440 to get \frac{37}{192}\sqrt{21}.

$$\frac{37}{192}\sqrt{21}$$