Solve sqrt{1390/182-113^2/2*182^2}

$$\sqrt{ \frac{ 1390 }{ 182 } - \frac{ { 113 }^{ 2 } }{ 2 \times { 182 }^{ 2 } } }$$

Evaluate

$$\frac{3\sqrt{109598}}{364}\approx 2.728482581$$

Solution Steps

Reduce the fraction \frac{1390}{182} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{695}{91}-\frac{113^{2}}{2\times 182^{2}}}$$

Calculate 113 to the power of 2 and get 12769.

$$\sqrt{\frac{695}{91}-\frac{12769}{2\times 182^{2}}}$$

Calculate 182 to the power of 2 and get 33124.

$$\sqrt{\frac{695}{91}-\frac{12769}{2\times 33124}}$$

Multiply 2 and 33124 to get 66248.

$$\sqrt{\frac{695}{91}-\frac{12769}{66248}}$$

Least common multiple of 91 and 66248 is 66248. Convert \frac{695}{91} and \frac{12769}{66248} to fractions with denominator 66248.

$$\sqrt{\frac{505960}{66248}-\frac{12769}{66248}}$$

Since \frac{505960}{66248} and \frac{12769}{66248} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{505960-12769}{66248}}$$

Subtract 12769 from 505960 to get 493191.

$$\sqrt{\frac{493191}{66248}}$$

Rewrite the square root of the division \sqrt{\frac{493191}{66248}} as the division of square roots \frac{\sqrt{493191}}{\sqrt{66248}}.

$$\frac{\sqrt{493191}}{\sqrt{66248}}$$

Factor 493191=3^{2}\times 54799. Rewrite the square root of the product \sqrt{3^{2}\times 54799} as the product of square roots \sqrt{3^{2}}\sqrt{54799}. Take the square root of 3^{2}.

$$\frac{3\sqrt{54799}}{\sqrt{66248}}$$

Factor 66248=182^{2}\times 2. Rewrite the square root of the product \sqrt{182^{2}\times 2} as the product of square roots \sqrt{182^{2}}\sqrt{2}. Take the square root of 182^{2}.

$$\frac{3\sqrt{54799}}{182\sqrt{2}}$$

Rationalize the denominator of \frac{3\sqrt{54799}}{182\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\sqrt{54799}\sqrt{2}}{182\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{54799}\sqrt{2}}{182\times 2}$$

To multiply \sqrt{54799} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{109598}}{182\times 2}$$

Multiply 182 and 2 to get 364.

$$\frac{3\sqrt{109598}}{364}$$