Solve -32a^8/-4a^3

$$\frac { - 32 a ^ { 8 } } { - 4 a ^ { 3 } }$$

Evaluate

$$8a^{5}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(-32a^{8}\right)^{1}\times \frac{1}{-4a^{3}}$$

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

$$\left(-32\right)^{1}\left(a^{8}\right)^{1}\times \frac{1}{-4}\times \frac{1}{a^{3}}$$

Use the Commutative Property of Multiplication.

$$\left(-32\right)^{1}\times \frac{1}{-4}\left(a^{8}\right)^{1}\times \frac{1}{a^{3}}$$

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

$$\left(-32\right)^{1}\times \frac{1}{-4}a^{8}a^{3\left(-1\right)}$$

Multiply 3 times -1.

$$\left(-32\right)^{1}\times \frac{1}{-4}a^{8}a^{-3}$$

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

$$\left(-32\right)^{1}\times \frac{1}{-4}a^{8-3}$$

Add the exponents 8 and -3.

$$\left(-32\right)^{1}\times \frac{1}{-4}a^{5}$$

Raise -32 to the power 1.

$$-32\times \frac{1}{-4}a^{5}$$

Raise -4 to the power -1.

$$-32\left(-\frac{1}{4}\right)a^{5}$$

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

$$8a^{5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{\left(-32\right)^{1}a^{8}}{\left(-4\right)^{1}a^{3}}$$

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

$$\frac{\left(-32\right)^{1}a^{8-3}}{\left(-4\right)^{1}}$$

Subtract 3 from 8.

$$\frac{\left(-32\right)^{1}a^{5}}{\left(-4\right)^{1}}$$

Divide -32 by -4.

$$8a^{5}$$

Differentiate w.r.t. a

$$40a^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-\frac{32}{-4}\right)a^{8-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(8a^{5})$$

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}.

$$5\times 8a^{5-1}$$

Do the arithmetic.

$$40a^{4}$$