Solve 3ab^-2/3^-1a^-7b^4

$$\frac { 3 a b ^ { - 2 } } { 3 ^ { - 1 } a ^ { - 7 } b ^ { 4 } }$$

Evaluate

$$\frac{9a^{8}}{b^{6}}$$

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$\frac{9a^{8}}{b^{6}}$$

Differentiate w.r.t. a

$$\frac{72a^{7}}{b^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{b^{2}\times \frac{b^{4}}{3}}a^{1-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{9}{b^{6}}a^{8})$$

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}.

$$8\times \frac{9}{b^{6}}a^{8-1}$$

Do the arithmetic.

$$\frac{72}{b^{6}}a^{7}$$