Solve frac{4a^{4}b^-5}{4^-1a^-4b^3}=

$$\frac { 4 a ^ { 4 } b ^ { - 5 } } { 4 ^ { - 1 } a ^ { - 4 } b ^ { 3 } } =$$

Evaluate

$$16\times \left(\frac{a}{b}\right)^{8}$$

Solution Steps

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

$$\frac{4^{2}b^{-5}a^{8}}{b^{3}}$$

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

$$\frac{4^{2}a^{8}}{b^{8}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16a^{8}}{b^{8}}$$

Differentiate w.r.t. a

$$\frac{128a^{7}}{b^{8}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{4}{b^{5}\times \frac{b^{3}}{4}}a^{4-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{16}{b^{8}}a^{8})$$

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}.

$$8\times \frac{16}{b^{8}}a^{8-1}$$

Do the arithmetic.

$$\frac{128}{b^{8}}a^{7}$$