Solve ((-1/4)^{3}:(-1/4)+(2-1/2)^{4}(1-5/9)^2):[(frac{17}{2})^4:(frac{17}{2})^3+frac{3}{2}*(-1)^2-1+frac{1}{4}]*frac{37}{2}

$$( ( - \frac { 1 } { 4 } ) ^ { 3 } : ( - \frac { 1 } { 4 } ) + ( 2 - \frac { 1 } { 2 } ) ^ { 4 } ( 1 - \frac { 5 } { 9 } ) ^ { 2 } ) : [ ( \frac { 17 } { 2 } ) ^ { 4 } : ( \frac { 17 } { 2 } ) ^ { 3 } + \frac { 3 } { 2 } \cdot ( - 1 ) ^ { 2 } - 1 + \frac { 1 } { 4 } ] \cdot \frac { 37 } { 2 }$$

Evaluate

$$\frac{17}{8}=2.125$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 3 to get 2.

$$\frac{\left(-\frac{1}{4}\right)^{2}+\left(2-\frac{1}{2}\right)^{4}\left(1-\frac{5}{9}\right)^{2}}{\frac{\left(\frac{17}{2}\right)^{4}}{\left(\frac{17}{2}\right)^{3}}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 4 to get 1.

$$\frac{\left(-\frac{1}{4}\right)^{2}+\left(2-\frac{1}{2}\right)^{4}\left(1-\frac{5}{9}\right)^{2}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

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

$$\frac{\frac{1}{16}+\left(2-\frac{1}{2}\right)^{4}\left(1-\frac{5}{9}\right)^{2}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Subtract \frac{1}{2} from 2 to get \frac{3}{2}.

$$\frac{\frac{1}{16}+\left(\frac{3}{2}\right)^{4}\left(1-\frac{5}{9}\right)^{2}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Calculate \frac{3}{2} to the power of 4 and get \frac{81}{16}.

$$\frac{\frac{1}{16}+\frac{81}{16}\left(1-\frac{5}{9}\right)^{2}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Subtract \frac{5}{9} from 1 to get \frac{4}{9}.

$$\frac{\frac{1}{16}+\frac{81}{16}\times \left(\frac{4}{9}\right)^{2}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Calculate \frac{4}{9} to the power of 2 and get \frac{16}{81}.

$$\frac{\frac{1}{16}+\frac{81}{16}\times \frac{16}{81}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Multiply \frac{81}{16} and \frac{16}{81} to get 1.

$$\frac{\frac{1}{16}+1}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Add \frac{1}{16} and 1 to get \frac{17}{16}.

$$\frac{\frac{17}{16}}{\left(\frac{17}{2}\right)^{1}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Calculate \frac{17}{2} to the power of 1 and get \frac{17}{2}.

$$\frac{\frac{17}{16}}{\frac{17}{2}+\frac{3}{2}\left(-1\right)^{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Calculate -1 to the power of 2 and get 1.

$$\frac{\frac{17}{16}}{\frac{17}{2}+\frac{3}{2}\times 1-1+\frac{1}{4}}\times \frac{37}{2}$$

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

$$\frac{\frac{17}{16}}{\frac{17}{2}+\frac{3}{2}-1+\frac{1}{4}}\times \frac{37}{2}$$

Add \frac{17}{2} and \frac{3}{2} to get 10.

$$\frac{\frac{17}{16}}{10-1+\frac{1}{4}}\times \frac{37}{2}$$

Subtract 1 from 10 to get 9.

$$\frac{\frac{17}{16}}{9+\frac{1}{4}}\times \frac{37}{2}$$

Add 9 and \frac{1}{4} to get \frac{37}{4}.

$$\frac{\frac{17}{16}}{\frac{37}{4}}\times \frac{37}{2}$$

Divide \frac{17}{16} by \frac{37}{4} by multiplying \frac{17}{16} by the reciprocal of \frac{37}{4}.

$$\frac{17}{16}\times \frac{4}{37}\times \frac{37}{2}$$

Multiply \frac{17}{16} and \frac{4}{37} to get \frac{17}{148}.

$$\frac{17}{148}\times \frac{37}{2}$$

Multiply \frac{17}{148} and \frac{37}{2} to get \frac{17}{8}.

$$\frac{17}{8}$$

Factor

$$\frac{17}{2 ^ {3}} = 2\frac{1}{8} = 2.125$$