Solve 44.5-28.1/sqrt{frac{8.68^2{10}+8.54^2/10}}

$$\frac{ 44.5-28.1 }{ \sqrt{ \frac{ { 8.68 }^{ 2 } }{ 10 } + \frac{ { 8.54 }^{ 2 } }{ 10 } } }$$

Evaluate

$$\frac{820\sqrt{3026}}{10591}\approx 4.259036345$$

Solution Steps

Subtract 28.1 from 44.5 to get 16.4.

$$\frac{16.4}{\sqrt{\frac{8.68^{2}}{10}+\frac{8.54^{2}}{10}}}$$

Calculate 8.68 to the power of 2 and get 75.3424.

$$\frac{16.4}{\sqrt{\frac{75.3424}{10}+\frac{8.54^{2}}{10}}}$$

Expand \frac{75.3424}{10} by multiplying both numerator and the denominator by 10000.

$$\frac{16.4}{\sqrt{\frac{753424}{100000}+\frac{8.54^{2}}{10}}}$$

Reduce the fraction \frac{753424}{100000} to lowest terms by extracting and canceling out 16.

$$\frac{16.4}{\sqrt{\frac{47089}{6250}+\frac{8.54^{2}}{10}}}$$

Calculate 8.54 to the power of 2 and get 72.9316.

$$\frac{16.4}{\sqrt{\frac{47089}{6250}+\frac{72.9316}{10}}}$$

Expand \frac{72.9316}{10} by multiplying both numerator and the denominator by 10000.

$$\frac{16.4}{\sqrt{\frac{47089}{6250}+\frac{729316}{100000}}}$$

Reduce the fraction \frac{729316}{100000} to lowest terms by extracting and canceling out 4.

$$\frac{16.4}{\sqrt{\frac{47089}{6250}+\frac{182329}{25000}}}$$

Least common multiple of 6250 and 25000 is 25000. Convert \frac{47089}{6250} and \frac{182329}{25000} to fractions with denominator 25000.

$$\frac{16.4}{\sqrt{\frac{188356}{25000}+\frac{182329}{25000}}}$$

Since \frac{188356}{25000} and \frac{182329}{25000} have the same denominator, add them by adding their numerators.

$$\frac{16.4}{\sqrt{\frac{188356+182329}{25000}}}$$

Add 188356 and 182329 to get 370685.

$$\frac{16.4}{\sqrt{\frac{370685}{25000}}}$$

Reduce the fraction \frac{370685}{25000} to lowest terms by extracting and canceling out 5.

$$\frac{16.4}{\sqrt{\frac{74137}{5000}}}$$

Rewrite the square root of the division \sqrt{\frac{74137}{5000}} as the division of square roots \frac{\sqrt{74137}}{\sqrt{5000}}.

$$\frac{16.4}{\frac{\sqrt{74137}}{\sqrt{5000}}}$$

Factor 74137=7^{2}\times 1513. Rewrite the square root of the product \sqrt{7^{2}\times 1513} as the product of square roots \sqrt{7^{2}}\sqrt{1513}. Take the square root of 7^{2}.

$$\frac{16.4}{\frac{7\sqrt{1513}}{\sqrt{5000}}}$$

Factor 5000=50^{2}\times 2. Rewrite the square root of the product \sqrt{50^{2}\times 2} as the product of square roots \sqrt{50^{2}}\sqrt{2}. Take the square root of 50^{2}.

$$\frac{16.4}{\frac{7\sqrt{1513}}{50\sqrt{2}}}$$

Rationalize the denominator of \frac{7\sqrt{1513}}{50\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{16.4}{\frac{7\sqrt{1513}\sqrt{2}}{50\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{16.4}{\frac{7\sqrt{1513}\sqrt{2}}{50\times 2}}$$

To multiply \sqrt{1513} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{16.4}{\frac{7\sqrt{3026}}{50\times 2}}$$

Multiply 50 and 2 to get 100.

$$\frac{16.4}{\frac{7\sqrt{3026}}{100}}$$

Divide 16.4 by \frac{7\sqrt{3026}}{100} by multiplying 16.4 by the reciprocal of \frac{7\sqrt{3026}}{100}.

$$\frac{16.4\times 100}{7\sqrt{3026}}$$

Rationalize the denominator of \frac{16.4\times 100}{7\sqrt{3026}} by multiplying numerator and denominator by \sqrt{3026}.

$$\frac{16.4\times 100\sqrt{3026}}{7\left(\sqrt{3026}\right)^{2}}$$

The square of \sqrt{3026} is 3026.

$$\frac{16.4\times 100\sqrt{3026}}{7\times 3026}$$

Multiply 16.4 and 100 to get 1640.

$$\frac{1640\sqrt{3026}}{7\times 3026}$$

Multiply 7 and 3026 to get 21182.

$$\frac{1640\sqrt{3026}}{21182}$$

Divide 1640\sqrt{3026} by 21182 to get \frac{820}{10591}\sqrt{3026}.

$$\frac{820}{10591}\sqrt{3026}$$