Solve frac{27a^{-3}b^-2}{3^-4a^2b^-3}

$$\frac { 27 a ^ { - 3 } b ^ { - 2 } } { 3 ^ { - 4 } a ^ { 2 } b ^ { - 3 } }$$

Evaluate

$$\frac{2187b}{a^{5}}$$

Solution Steps

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

$$\frac{27a^{-3}b^{1}}{3^{-4}a^{2}}$$

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

$$\frac{27b^{1}}{3^{-4}a^{5}}$$

Calculate b to the power of 1 and get b.

$$\frac{27b}{3^{-4}a^{5}}$$

Calculate 3 to the power of -4 and get \frac{1}{81}.

$$\frac{27b}{\frac{1}{81}a^{5}}$$

Differentiate w.r.t. a

$$-\frac{10935b}{a^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{27}{b^{2}\times \frac{1}{81b^{3}}}a^{-3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2187ba^{-5})$$

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}.

$$-5\times 2187ba^{-5-1}$$

Do the arithmetic.

$$\left(-10935b\right)a^{-6}$$