Solve {left(frac{1/2right)}^2-2{left(2/3right)}^2}{1/4}+{left(frac{1/4right)}^2+3^2}{frac{2}{3}}

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

Evaluate

$$\frac{3179}{288}\approx 11.038194444$$

Solution Steps

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

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

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{1}{4}-2\times \frac{4}{9}}{\frac{1}{4}}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Express 2\times \frac{4}{9} as a single fraction.

$$\frac{\frac{1}{4}-\frac{2\times 4}{9}}{\frac{1}{4}}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Multiply 2 and 4 to get 8.

$$\frac{\frac{1}{4}-\frac{8}{9}}{\frac{1}{4}}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Least common multiple of 4 and 9 is 36. Convert \frac{1}{4} and \frac{8}{9} to fractions with denominator 36.

$$\frac{\frac{9}{36}-\frac{32}{36}}{\frac{1}{4}}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Since \frac{9}{36} and \frac{32}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{9-32}{36}}{\frac{1}{4}}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Subtract 32 from 9 to get -23.

$$\frac{-\frac{23}{36}}{\frac{1}{4}}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Divide -\frac{23}{36} by \frac{1}{4} by multiplying -\frac{23}{36} by the reciprocal of \frac{1}{4}.

$$-\frac{23}{36}\times 4+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Express -\frac{23}{36}\times 4 as a single fraction.

$$\frac{-23\times 4}{36}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Multiply -23 and 4 to get -92.

$$\frac{-92}{36}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Reduce the fraction \frac{-92}{36} to lowest terms by extracting and canceling out 4.

$$-\frac{23}{9}+\frac{\left(\frac{1}{4}\right)^{2}+3^{2}}{\frac{2}{3}}$$

Calculate \frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$-\frac{23}{9}+\frac{\frac{1}{16}+3^{2}}{\frac{2}{3}}$$

Calculate 3 to the power of 2 and get 9.

$$-\frac{23}{9}+\frac{\frac{1}{16}+9}{\frac{2}{3}}$$

Convert 9 to fraction \frac{144}{16}.

$$-\frac{23}{9}+\frac{\frac{1}{16}+\frac{144}{16}}{\frac{2}{3}}$$

Since \frac{1}{16} and \frac{144}{16} have the same denominator, add them by adding their numerators.

$$-\frac{23}{9}+\frac{\frac{1+144}{16}}{\frac{2}{3}}$$

Add 1 and 144 to get 145.

$$-\frac{23}{9}+\frac{\frac{145}{16}}{\frac{2}{3}}$$

Divide \frac{145}{16} by \frac{2}{3} by multiplying \frac{145}{16} by the reciprocal of \frac{2}{3}.

$$-\frac{23}{9}+\frac{145}{16}\times \frac{3}{2}$$

Multiply \frac{145}{16} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{23}{9}+\frac{145\times 3}{16\times 2}$$

Do the multiplications in the fraction \frac{145\times 3}{16\times 2}.

$$-\frac{23}{9}+\frac{435}{32}$$

Least common multiple of 9 and 32 is 288. Convert -\frac{23}{9} and \frac{435}{32} to fractions with denominator 288.

$$-\frac{736}{288}+\frac{3915}{288}$$

Since -\frac{736}{288} and \frac{3915}{288} have the same denominator, add them by adding their numerators.

$$\frac{-736+3915}{288}$$

Add -736 and 3915 to get 3179.

$$\frac{3179}{288}$$

Factor

$$\frac{11 \cdot 17 ^ {2}}{2 ^ {5} \cdot 3 ^ {2}} = 11\frac{11}{288} = 11.038194444444445$$