Solve frac{(-12x^{4}y^{5}z^{2})*(-8x^2y^3z)}{16x^3y^4z}

$$\frac { ( - 12 x ^ { 4 } y ^ { 5 } z ^ { 2 } ) \times ( - 8 x ^ { 2 } y ^ { 3 } z ) } { 16 x ^ { 3 } y ^ { 4 } z }$$

Evaluate

$$6z^{2}x^{3}y^{4}$$

Solution Steps

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

$$\frac{-12x^{6}y^{5}z^{2}\left(-8\right)y^{3}z}{16x^{3}y^{4}z}$$

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$\frac{-12x^{6}y^{8}z^{2}\left(-8\right)z}{16x^{3}y^{4}z}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{-12x^{6}y^{8}z^{3}\left(-8\right)}{16x^{3}y^{4}z}$$

Cancel out 4\times 4zx^{3}y^{4} in both numerator and denominator.

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

Multiply -3 and -2 to get 6.

$$6z^{2}x^{3}y^{4}$$

Differentiate w.r.t. x

$$18y^{4}\left(xz\right)^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{96x^{2}z^{3}y^{8}}{16zy^{4}}x^{4-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{2}z^{2}y^{4}x^{1})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$6x^{2}z^{2}y^{4}x^{1-1}$$

Do the arithmetic.

$$6x^{2}z^{2}y^{4}x^{0}$$

For any term t except 0, t^{0}=1.

$$6x^{2}z^{2}y^{4}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$6x^{2}z^{2}y^{4}$$