Solve x^{-2}y^{0}*y^3*(-x^5y^3)^4

$$x ^ { - 2 } y ^ { 0 } \cdot y ^ { 3 } \cdot ( - x ^ { 5 } y ^ { 3 } ) ^ { 4 }$$

Evaluate

$$y^{15}x^{18}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 0 and 3 to get 3.

$$x^{-2}y^{3}\left(\left(-x^{5}\right)y^{3}\right)^{4}$$

Expand \left(\left(-x^{5}\right)y^{3}\right)^{4}.

$$x^{-2}y^{3}\left(-x^{5}\right)^{4}\left(y^{3}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$x^{-2}y^{3}\left(-x^{5}\right)^{4}y^{12}$$

To multiply powers of the same base, add their exponents. Add 3 and 12 to get 15.

$$x^{-2}y^{15}\left(-x^{5}\right)^{4}$$

Expand \left(-x^{5}\right)^{4}.

$$x^{-2}y^{15}\left(-1\right)^{4}\left(x^{5}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$x^{-2}y^{15}\left(-1\right)^{4}x^{20}$$

Calculate -1 to the power of 4 and get 1.

$$x^{-2}y^{15}\times 1x^{20}$$

To multiply powers of the same base, add their exponents. Add -2 and 20 to get 18.

$$x^{18}y^{15}\times 1$$

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

$$x^{18}y^{15}$$

Differentiate w.r.t. x

$$18y^{15}x^{17}$$