Solve sqrt{(11/4*8/11)^2:[(23/12-3/2):frac{5}{4}]^2}-sqrt{10+[(frac{1}{2})^2:frac{1}{2}+frac{12}{13}*(frac{5}{4}-frac{1}{6})]:frac{8}{3}}

$$\sqrt { ( \frac { 11 } { 4 } \times \frac { 8 } { 11 } ) ^ { 2 } : [ ( \frac { 23 } { 12 } - \frac { 3 } { 2 } ) : \frac { 5 } { 4 } ] ^ { 2 } } - \sqrt { 10 + [ ( \frac { 1 } { 2 } ) ^ { 2 } : \frac { 1 } { 2 } + \frac { 12 } { 13 } \times ( \frac { 5 } { 4 } - \frac { 1 } { 6 } ) ] : \frac { 8 } { 3 } }$$

Evaluate

$$\frac{11}{4}=2.75$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 2 to get 1.

$$\sqrt{\frac{\left(\frac{11}{4}\times \frac{8}{11}\right)^{2}}{\left(\frac{\frac{23}{12}-\frac{3}{2}}{\frac{5}{4}}\right)^{2}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Multiply \frac{11}{4} and \frac{8}{11} to get 2.

$$\sqrt{\frac{2^{2}}{\left(\frac{\frac{23}{12}-\frac{3}{2}}{\frac{5}{4}}\right)^{2}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{\frac{4}{\left(\frac{\frac{23}{12}-\frac{3}{2}}{\frac{5}{4}}\right)^{2}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Subtract \frac{3}{2} from \frac{23}{12} to get \frac{5}{12}.

$$\sqrt{\frac{4}{\left(\frac{\frac{5}{12}}{\frac{5}{4}}\right)^{2}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Divide \frac{5}{12} by \frac{5}{4} by multiplying \frac{5}{12} by the reciprocal of \frac{5}{4}.

$$\sqrt{\frac{4}{\left(\frac{5}{12}\times \frac{4}{5}\right)^{2}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Multiply \frac{5}{12} and \frac{4}{5} to get \frac{1}{3}.

$$\sqrt{\frac{4}{\left(\frac{1}{3}\right)^{2}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\sqrt{\frac{4}{\frac{1}{9}}}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Divide 4 by \frac{1}{9} by multiplying 4 by the reciprocal of \frac{1}{9}.

$$\sqrt{4\times 9}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Multiply 4 and 9 to get 36.

$$\sqrt{36}-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Calculate the square root of 36 and get 6.

$$6-\sqrt{10+\frac{\left(\frac{1}{2}\right)^{1}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

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

$$6-\sqrt{10+\frac{\frac{1}{2}+\frac{12}{13}\left(\frac{5}{4}-\frac{1}{6}\right)}{\frac{8}{3}}}$$

Subtract \frac{1}{6} from \frac{5}{4} to get \frac{13}{12}.

$$6-\sqrt{10+\frac{\frac{1}{2}+\frac{12}{13}\times \frac{13}{12}}{\frac{8}{3}}}$$

Multiply \frac{12}{13} and \frac{13}{12} to get 1.

$$6-\sqrt{10+\frac{\frac{1}{2}+1}{\frac{8}{3}}}$$

Add \frac{1}{2} and 1 to get \frac{3}{2}.

$$6-\sqrt{10+\frac{\frac{3}{2}}{\frac{8}{3}}}$$

Divide \frac{3}{2} by \frac{8}{3} by multiplying \frac{3}{2} by the reciprocal of \frac{8}{3}.

$$6-\sqrt{10+\frac{3}{2}\times \frac{3}{8}}$$

Multiply \frac{3}{2} and \frac{3}{8} to get \frac{9}{16}.

$$6-\sqrt{10+\frac{9}{16}}$$

Add 10 and \frac{9}{16} to get \frac{169}{16}.

$$6-\sqrt{\frac{169}{16}}$$

Rewrite the square root of the division \frac{169}{16} as the division of square roots \frac{\sqrt{169}}{\sqrt{16}}. Take the square root of both numerator and denominator.

$$6-\frac{13}{4}$$

Subtract \frac{13}{4} from 6 to get \frac{11}{4}.

$$\frac{11}{4}$$

Factor

$$\frac{11}{2 ^ {2}} = 2\frac{3}{4} = 2.75$$