Solve -32x^12/4x^8

$$\frac{ -32 { x }^{ 12 } }{ 4 { x }^{ 8 } }$$

Evaluate

$$-8x^{4}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(-32x^{12}\right)^{1}\times \frac{1}{4x^{8}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\left(-32\right)^{1}\left(x^{12}\right)^{1}\times \frac{1}{4}\times \frac{1}{x^{8}}$$

Use the Commutative Property of Multiplication.

$$\left(-32\right)^{1}\times \frac{1}{4}\left(x^{12}\right)^{1}\times \frac{1}{x^{8}}$$

To raise a power to another power, multiply the exponents.

$$\left(-32\right)^{1}\times \frac{1}{4}x^{12}x^{8\left(-1\right)}$$

Multiply 8 times -1.

$$\left(-32\right)^{1}\times \frac{1}{4}x^{12}x^{-8}$$

To multiply powers of the same base, add their exponents.

$$\left(-32\right)^{1}\times \frac{1}{4}x^{12-8}$$

Add the exponents 12 and -8.

$$\left(-32\right)^{1}\times \frac{1}{4}x^{4}$$

Raise -32 to the power 1.

$$-32\times \frac{1}{4}x^{4}$$

Multiply -32 times \frac{1}{4}.

$$-8x^{4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{\left(-32\right)^{1}x^{12}}{4^{1}x^{8}}$$

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

$$\frac{\left(-32\right)^{1}x^{12-8}}{4^{1}}$$

Subtract 8 from 12.

$$\frac{\left(-32\right)^{1}x^{4}}{4^{1}}$$

Divide -32 by 4.

$$-8x^{4}$$

Differentiate w.r.t. x

$$-32x^{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{32}{4}\right)x^{12-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-8x^{4})$$

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}.

$$4\left(-8\right)x^{4-1}$$

Do the arithmetic.

$$-32x^{3}$$