Solve (-3/2a^{4}b^{5}c^{8}):(-9/4ab^2c^2):(-3/4a^2bc):(-abc^2)

$$( - \frac { 3 } { 2 } a ^ { 4 } b ^ { 5 } c ^ { 8 } ) : ( - \frac { 9 } { 4 } a b ^ { 2 } c ^ { 2 } ) : ( - \frac { 3 } { 4 } a ^ { 2 } b c ) : ( - a b c ^ { 2 } )$$

Evaluate

$$\frac{8bc^{3}}{9}$$

Solution Steps

Express \frac{\frac{-\frac{3}{2}a^{4}b^{5}c^{8}}{-\frac{9}{4}ab^{2}c^{2}}}{-\frac{3}{4}a^{2}bc} as a single fraction.

$$\frac{\frac{-\frac{3}{2}a^{4}b^{5}c^{8}}{-\frac{9}{4}ab^{2}c^{2}\left(-\frac{3}{4}\right)a^{2}bc}}{\left(-a\right)bc^{2}}$$

Cancel out abca^{2}b^{2}c^{2} in both numerator and denominator.

$$\frac{\frac{-\frac{3}{2}ab^{2}c^{5}}{-\frac{9}{4}\left(-\frac{3}{4}\right)}}{\left(-a\right)bc^{2}}$$

Multiply -\frac{9}{4} and -\frac{3}{4} to get \frac{27}{16}.

$$\frac{\frac{-\frac{3}{2}ab^{2}c^{5}}{\frac{27}{16}}}{\left(-a\right)bc^{2}}$$

Divide -\frac{3}{2}ab^{2}c^{5} by \frac{27}{16} by multiplying -\frac{3}{2}ab^{2}c^{5} by the reciprocal of \frac{27}{16}.

$$\frac{\frac{-\frac{3}{2}ab^{2}c^{5}\times 16}{27}}{\left(-a\right)bc^{2}}$$

Multiply -\frac{3}{2} and 16 to get -24.

$$\frac{\frac{-24ab^{2}c^{5}}{27}}{\left(-a\right)bc^{2}}$$

Divide -24ab^{2}c^{5} by 27 to get -\frac{8}{9}ab^{2}c^{5}.

$$\frac{-\frac{8}{9}ab^{2}c^{5}}{\left(-a\right)bc^{2}}$$

Cancel out bc^{2} in both numerator and denominator.

$$\frac{-\frac{8}{9}abc^{3}}{-a}$$

Cancel out a in both numerator and denominator.

$$\frac{-\frac{8}{9}bc^{3}}{-1}$$

Anything divided by -1 gives its opposite.

$$\frac{8}{9}bc^{3}$$

Differentiate w.r.t. b

$$\frac{8c^{3}}{9}$$