Solve 1000/sqrt{frac{500{72}+2500/20.17}}

$$\frac{ 1000 }{ \sqrt{ \frac{ 500 }{ 72 } + \frac{ 2500 }{ 20.17 } } }$$

Evaluate

$$\frac{120\sqrt{766802890}}{38017}\approx 87.406810652$$

Solution Steps

Reduce the fraction \frac{500}{72} to lowest terms by extracting and canceling out 4.

$$\frac{1000}{\sqrt{\frac{125}{18}+\frac{2500}{20.17}}}$$

Expand \frac{2500}{20.17} by multiplying both numerator and the denominator by 100.

$$\frac{1000}{\sqrt{\frac{125}{18}+\frac{250000}{2017}}}$$

Least common multiple of 18 and 2017 is 36306. Convert \frac{125}{18} and \frac{250000}{2017} to fractions with denominator 36306.

$$\frac{1000}{\sqrt{\frac{252125}{36306}+\frac{4500000}{36306}}}$$

Since \frac{252125}{36306} and \frac{4500000}{36306} have the same denominator, add them by adding their numerators.

$$\frac{1000}{\sqrt{\frac{252125+4500000}{36306}}}$$

Add 252125 and 4500000 to get 4752125.

$$\frac{1000}{\sqrt{\frac{4752125}{36306}}}$$

Rewrite the square root of the division \sqrt{\frac{4752125}{36306}} as the division of square roots \frac{\sqrt{4752125}}{\sqrt{36306}}.

$$\frac{1000}{\frac{\sqrt{4752125}}{\sqrt{36306}}}$$

Factor 4752125=5^{2}\times 190085. Rewrite the square root of the product \sqrt{5^{2}\times 190085} as the product of square roots \sqrt{5^{2}}\sqrt{190085}. Take the square root of 5^{2}.

$$\frac{1000}{\frac{5\sqrt{190085}}{\sqrt{36306}}}$$

Factor 36306=3^{2}\times 4034. Rewrite the square root of the product \sqrt{3^{2}\times 4034} as the product of square roots \sqrt{3^{2}}\sqrt{4034}. Take the square root of 3^{2}.

$$\frac{1000}{\frac{5\sqrt{190085}}{3\sqrt{4034}}}$$

Rationalize the denominator of \frac{5\sqrt{190085}}{3\sqrt{4034}} by multiplying numerator and denominator by \sqrt{4034}.

$$\frac{1000}{\frac{5\sqrt{190085}\sqrt{4034}}{3\left(\sqrt{4034}\right)^{2}}}$$

The square of \sqrt{4034} is 4034.

$$\frac{1000}{\frac{5\sqrt{190085}\sqrt{4034}}{3\times 4034}}$$

To multiply \sqrt{190085} and \sqrt{4034}, multiply the numbers under the square root.

$$\frac{1000}{\frac{5\sqrt{766802890}}{3\times 4034}}$$

Multiply 3 and 4034 to get 12102.

$$\frac{1000}{\frac{5\sqrt{766802890}}{12102}}$$

Divide 1000 by \frac{5\sqrt{766802890}}{12102} by multiplying 1000 by the reciprocal of \frac{5\sqrt{766802890}}{12102}.

$$\frac{1000\times 12102}{5\sqrt{766802890}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{200\times 12102}{\sqrt{766802890}}$$

Rationalize the denominator of \frac{200\times 12102}{\sqrt{766802890}} by multiplying numerator and denominator by \sqrt{766802890}.

$$\frac{200\times 12102\sqrt{766802890}}{\left(\sqrt{766802890}\right)^{2}}$$

The square of \sqrt{766802890} is 766802890.

$$\frac{200\times 12102\sqrt{766802890}}{766802890}$$

Multiply 200 and 12102 to get 2420400.

$$\frac{2420400\sqrt{766802890}}{766802890}$$

Divide 2420400\sqrt{766802890} by 766802890 to get \frac{120}{38017}\sqrt{766802890}.

$$\frac{120}{38017}\sqrt{766802890}$$