Solve sqrt{8/11*11/4+(1/2)^2/frac{2{3}+(frac{11}{2})^2}}

$$\sqrt { \frac { \frac { 8 } { 11 } \times \frac { 11 } { 4 } + ( \frac { 1 } { 2 } ) ^ { 2 } } { \frac { 2 } { 3 } + ( \frac { 11 } { 2 } ) ^ { 2 } } }$$

Evaluate

$$\frac{3\sqrt{1113}}{371}\approx 0.269770792$$

Solution Steps

Multiply \frac{8}{11} times \frac{11}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{8\times 11}{11\times 4}+\left(\frac{1}{2}\right)^{2}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Cancel out 11 in both numerator and denominator.

$$\sqrt{\frac{\frac{8}{4}+\left(\frac{1}{2}\right)^{2}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Divide 8 by 4 to get 2.

$$\sqrt{\frac{2+\left(\frac{1}{2}\right)^{2}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\sqrt{\frac{2+\frac{1}{4}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Convert 2 to fraction \frac{8}{4}.

$$\sqrt{\frac{\frac{8}{4}+\frac{1}{4}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Since \frac{8}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{8+1}{4}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Add 8 and 1 to get 9.

$$\sqrt{\frac{\frac{9}{4}}{\frac{2}{3}+\left(\frac{11}{2}\right)^{2}}}$$

Calculate \frac{11}{2} to the power of 2 and get \frac{121}{4}.

$$\sqrt{\frac{\frac{9}{4}}{\frac{2}{3}+\frac{121}{4}}}$$

Least common multiple of 3 and 4 is 12. Convert \frac{2}{3} and \frac{121}{4} to fractions with denominator 12.

$$\sqrt{\frac{\frac{9}{4}}{\frac{8}{12}+\frac{363}{12}}}$$

Since \frac{8}{12} and \frac{363}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{9}{4}}{\frac{8+363}{12}}}$$

Add 8 and 363 to get 371.

$$\sqrt{\frac{\frac{9}{4}}{\frac{371}{12}}}$$

Divide \frac{9}{4} by \frac{371}{12} by multiplying \frac{9}{4} by the reciprocal of \frac{371}{12}.

$$\sqrt{\frac{9}{4}\times \frac{12}{371}}$$

Multiply \frac{9}{4} times \frac{12}{371} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{9\times 12}{4\times 371}}$$

Do the multiplications in the fraction \frac{9\times 12}{4\times 371}.

$$\sqrt{\frac{108}{1484}}$$

Reduce the fraction \frac{108}{1484} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{27}{371}}$$

Rewrite the square root of the division \sqrt{\frac{27}{371}} as the division of square roots \frac{\sqrt{27}}{\sqrt{371}}.

$$\frac{\sqrt{27}}{\sqrt{371}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{3\sqrt{3}}{\sqrt{371}}$$

Rationalize the denominator of \frac{3\sqrt{3}}{\sqrt{371}} by multiplying numerator and denominator by \sqrt{371}.

$$\frac{3\sqrt{3}\sqrt{371}}{\left(\sqrt{371}\right)^{2}}$$

The square of \sqrt{371} is 371.

$$\frac{3\sqrt{3}\sqrt{371}}{371}$$

To multiply \sqrt{3} and \sqrt{371}, multiply the numbers under the square root.

$$\frac{3\sqrt{1113}}{371}$$