Solve frac{7^{3}x(-4)^{3}[(-4)^{2}]^5}{[(-4)^3]^4*7^3}

$$\frac { 7 ^ { 3 } x ( - 4 ) ^ { 3 } [ ( - 4 ) ^ { 2 } ] ^ { 5 } } { [ ( - 4 ) ^ { 3 } ] ^ { 4 } \times 7 ^ { 3 } }$$

Evaluate

$$-4x$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{7^{3}x\left(-4\right)^{3}\left(-4\right)^{10}}{\left(\left(-4\right)^{3}\right)^{4}\times 7^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 10 to get 13.

$$\frac{7^{3}x\left(-4\right)^{13}}{\left(\left(-4\right)^{3}\right)^{4}\times 7^{3}}$$

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

$$\frac{7^{3}x\left(-4\right)^{13}}{\left(-4\right)^{12}\times 7^{3}}$$

Cancel out \left(-4\right)^{12}\times 7^{3} in both numerator and denominator.

$$-4x$$

Differentiate w.r.t. x

$$-4$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7^{3}x\left(-4\right)^{3}\left(-4\right)^{10}}{\left(\left(-4\right)^{3}\right)^{4}\times 7^{3}})$$

To multiply powers of the same base, add their exponents. Add 3 and 10 to get 13.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7^{3}x\left(-4\right)^{13}}{\left(\left(-4\right)^{3}\right)^{4}\times 7^{3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7^{3}x\left(-4\right)^{13}}{\left(-4\right)^{12}\times 7^{3}})$$

Cancel out \left(-4\right)^{12}\times 7^{3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-4x)$$

The derivative of ax^{n} is nax^{n-1}.

$$-4x^{1-1}$$

Subtract 1 from 1.

$$-4x^{0}$$

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

$$-4$$