Solve 4a^6b^2/-8a^3b^2

$$\frac { 4 a ^ { 6 } b ^ { 2 } } { - 8 a ^ { 3 } b ^ { 2 } }$$

Evaluate

$$-\frac{a^{3}}{2}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}a^{6}b^{2}}{\left(-8\right)^{1}a^{3}b^{2}}$$

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

$$\frac{4^{1}}{\left(-8\right)^{1}}a^{6-3}b^{2-2}$$

Subtract 3 from 6.

$$\frac{4^{1}}{\left(-8\right)^{1}}a^{3}b^{2-2}$$

Subtract 2 from 2.

$$\frac{4^{1}}{\left(-8\right)^{1}}a^{3}b^{0}$$

For any number a except 0, a^{0}=1.

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

Reduce the fraction \frac{4}{-8} to lowest terms by extracting and canceling out 4.

$$-\frac{1}{2}a^{3}$$

Differentiate w.r.t. a

$$-\frac{3a^{2}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{4b^{2}}{-8b^{2}}a^{6-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{1}{2}a^{3})$$

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}.

$$3\left(-\frac{1}{2}\right)a^{3-1}$$

Do the arithmetic.

$$-\frac{3}{2}a^{2}$$