Solve (-6x^{12}y^{7}z^{4}):(-2x^{0}y^{5}z^3)*(-3/4xy^2z^3)^2

$$( - 6 x ^ { 12 } y ^ { 7 } z ^ { 4 } ) : ( - 2 x ^ { 0 } y ^ { 5 } z ^ { 3 } ) \cdot ( - \frac { 3 } { 4 } x y ^ { 2 } z ^ { 3 } ) ^ { 2 }$$

Evaluate

$$\frac{27y^{6}z^{7}x^{14}}{16}$$

Solution Steps

Cancel out 2z^{3}y^{5} in both numerator and denominator.

$$\frac{-3zy^{2}x^{12}}{-x^{0}}\left(-\frac{3}{4}xy^{2}z^{3}\right)^{2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{-3zy^{2}x^{12}}{-1}\left(-\frac{3}{4}xy^{2}z^{3}\right)^{2}$$

Anything divided by -1 gives its opposite.

$$3zy^{2}x^{12}\left(-\frac{3}{4}xy^{2}z^{3}\right)^{2}$$

Expand \left(-\frac{3}{4}xy^{2}z^{3}\right)^{2}.

$$3zy^{2}x^{12}\left(-\frac{3}{4}\right)^{2}x^{2}\left(y^{2}\right)^{2}\left(z^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$3zy^{2}x^{12}\left(-\frac{3}{4}\right)^{2}x^{2}y^{4}\left(z^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$3zy^{2}x^{12}\left(-\frac{3}{4}\right)^{2}x^{2}y^{4}z^{6}$$

Calculate -\frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$3zy^{2}x^{12}\times \frac{9}{16}x^{2}y^{4}z^{6}$$

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

$$\frac{27}{16}zy^{2}x^{12}x^{2}y^{4}z^{6}$$

To multiply powers of the same base, add their exponents. Add 12 and 2 to get 14.

$$\frac{27}{16}zy^{2}x^{14}y^{4}z^{6}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{27}{16}zy^{6}x^{14}z^{6}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{27}{16}z^{7}y^{6}x^{14}$$

Differentiate w.r.t. x

$$\frac{189y^{6}z^{7}x^{13}}{8}$$