Solve (-7/18)*(-4.5)+1/6*(-1)^2000/(-13frac{1{3})*(-1)^1009-(-33/4)+(-frac{5}{16})}+2frac{7}{8}

$$\frac { ( - \frac { 7 } { 18 } ) \times ( - 4.5 ) + \frac { 1 } { 6 } \times ( - 1 ) ^ { 2000 } } { ( - 13 \frac { 1 } { 3 } ) \times ( - 1 ) ^ { 1009 } - ( - 3 \frac { 3 } { 4 } ) + ( - \frac { 5 } { 16 } ) } + 2 \frac { 7 } { 8 }$$

Evaluate

$$\frac{837}{280}\approx 2.989285714$$

Solution Steps

Convert decimal number -4.5 to fraction -\frac{45}{10}. Reduce the fraction -\frac{45}{10} to lowest terms by extracting and canceling out 5.

$$\frac{-\frac{7}{18}\left(-\frac{9}{2}\right)+\frac{1}{6}\left(-1\right)^{2000}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Multiply -\frac{7}{18} times -\frac{9}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-7\left(-9\right)}{18\times 2}+\frac{1}{6}\left(-1\right)^{2000}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Do the multiplications in the fraction \frac{-7\left(-9\right)}{18\times 2}.

$$\frac{\frac{63}{36}+\frac{1}{6}\left(-1\right)^{2000}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Reduce the fraction \frac{63}{36} to lowest terms by extracting and canceling out 9.

$$\frac{\frac{7}{4}+\frac{1}{6}\left(-1\right)^{2000}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Calculate -1 to the power of 2000 and get 1.

$$\frac{\frac{7}{4}+\frac{1}{6}\times 1}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Multiply \frac{1}{6} and 1 to get \frac{1}{6}.

$$\frac{\frac{7}{4}+\frac{1}{6}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Least common multiple of 4 and 6 is 12. Convert \frac{7}{4} and \frac{1}{6} to fractions with denominator 12.

$$\frac{\frac{21}{12}+\frac{2}{12}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Since \frac{21}{12} and \frac{2}{12} have the same denominator, add them by adding their numerators.

$$\frac{\frac{21+2}{12}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Add 21 and 2 to get 23.

$$\frac{\frac{23}{12}}{\left(-\frac{13\times 3+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Multiply 13 and 3 to get 39.

$$\frac{\frac{23}{12}}{\left(-\frac{39+1}{3}\right)\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Add 39 and 1 to get 40.

$$\frac{\frac{23}{12}}{-\frac{40}{3}\left(-1\right)^{1009}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Calculate -1 to the power of 1009 and get -1.

$$\frac{\frac{23}{12}}{-\frac{40}{3}\left(-1\right)-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Multiply -\frac{40}{3} and -1 to get \frac{40}{3}.

$$\frac{\frac{23}{12}}{\frac{40}{3}-\left(-\frac{3\times 4+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Multiply 3 and 4 to get 12.

$$\frac{\frac{23}{12}}{\frac{40}{3}-\left(-\frac{12+3}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Add 12 and 3 to get 15.

$$\frac{\frac{23}{12}}{\frac{40}{3}-\left(-\frac{15}{4}\right)-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

The opposite of -\frac{15}{4} is \frac{15}{4}.

$$\frac{\frac{23}{12}}{\frac{40}{3}+\frac{15}{4}-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Least common multiple of 3 and 4 is 12. Convert \frac{40}{3} and \frac{15}{4} to fractions with denominator 12.

$$\frac{\frac{23}{12}}{\frac{160}{12}+\frac{45}{12}-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Since \frac{160}{12} and \frac{45}{12} have the same denominator, add them by adding their numerators.

$$\frac{\frac{23}{12}}{\frac{160+45}{12}-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Add 160 and 45 to get 205.

$$\frac{\frac{23}{12}}{\frac{205}{12}-\frac{5}{16}}+\frac{2\times 8+7}{8}$$

Least common multiple of 12 and 16 is 48. Convert \frac{205}{12} and \frac{5}{16} to fractions with denominator 48.

$$\frac{\frac{23}{12}}{\frac{820}{48}-\frac{15}{48}}+\frac{2\times 8+7}{8}$$

Since \frac{820}{48} and \frac{15}{48} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{23}{12}}{\frac{820-15}{48}}+\frac{2\times 8+7}{8}$$

Subtract 15 from 820 to get 805.

$$\frac{\frac{23}{12}}{\frac{805}{48}}+\frac{2\times 8+7}{8}$$

Divide \frac{23}{12} by \frac{805}{48} by multiplying \frac{23}{12} by the reciprocal of \frac{805}{48}.

$$\frac{23}{12}\times \frac{48}{805}+\frac{2\times 8+7}{8}$$

Multiply \frac{23}{12} times \frac{48}{805} by multiplying numerator times numerator and denominator times denominator.

$$\frac{23\times 48}{12\times 805}+\frac{2\times 8+7}{8}$$

Do the multiplications in the fraction \frac{23\times 48}{12\times 805}.

$$\frac{1104}{9660}+\frac{2\times 8+7}{8}$$

Reduce the fraction \frac{1104}{9660} to lowest terms by extracting and canceling out 276.

$$\frac{4}{35}+\frac{2\times 8+7}{8}$$

Multiply 2 and 8 to get 16.

$$\frac{4}{35}+\frac{16+7}{8}$$

Add 16 and 7 to get 23.

$$\frac{4}{35}+\frac{23}{8}$$

Least common multiple of 35 and 8 is 280. Convert \frac{4}{35} and \frac{23}{8} to fractions with denominator 280.

$$\frac{32}{280}+\frac{805}{280}$$

Since \frac{32}{280} and \frac{805}{280} have the same denominator, add them by adding their numerators.

$$\frac{32+805}{280}$$

Add 32 and 805 to get 837.

$$\frac{837}{280}$$

Factor

$$\frac{31 \cdot 3 ^ {3}}{5 \cdot 7 \cdot 2 ^ {3}} = 2\frac{277}{280} = 2.9892857142857143$$