Solve (+32a^8):(-4a^6)

$$( + 32 a ^ { 8 } ) : ( - 4 a ^ { 6 } )$$

Evaluate

$$-8a^{2}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(32a^{8}\right)^{1}\times \frac{1}{-4a^{6}}$$

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

$$32^{1}\left(a^{8}\right)^{1}\times \frac{1}{-4}\times \frac{1}{a^{6}}$$

Use the Commutative Property of Multiplication.

$$32^{1}\times \frac{1}{-4}\left(a^{8}\right)^{1}\times \frac{1}{a^{6}}$$

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

$$32^{1}\times \frac{1}{-4}a^{8}a^{6\left(-1\right)}$$

Multiply 6 times -1.

$$32^{1}\times \frac{1}{-4}a^{8}a^{-6}$$

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

$$32^{1}\times \frac{1}{-4}a^{8-6}$$

Add the exponents 8 and -6.

$$32^{1}\times \frac{1}{-4}a^{2}$$

Raise 32 to the power 1.

$$32\times \frac{1}{-4}a^{2}$$

Raise -4 to the power -1.

$$32\left(-\frac{1}{4}\right)a^{2}$$

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

$$-8a^{2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{32^{1}a^{8}}{\left(-4\right)^{1}a^{6}}$$

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

$$\frac{32^{1}a^{8-6}}{\left(-4\right)^{1}}$$

Subtract 6 from 8.

$$\frac{32^{1}a^{2}}{\left(-4\right)^{1}}$$

Divide 32 by -4.

$$-8a^{2}$$

Differentiate w.r.t. a

$$-16a$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{32}{-4}a^{8-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-8a^{2})$$

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}.

$$2\left(-8\right)a^{2-1}$$

Do the arithmetic.

$$-16a^{1}$$

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

$$-16a$$