Solve frac{2^{-3}a^{-1}b^-2}{4^-3a^2b^-1}=

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

Evaluate

$$\frac{8}{ba^{3}}$$

Solution Steps

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

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

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\frac{1}{8}}{4^{-3}b^{1}a^{3}}$$

Calculate 4 to the power of -3 and get \frac{1}{64}.

$$\frac{\frac{1}{8}}{\frac{1}{64}b^{1}a^{3}}$$

Calculate b to the power of 1 and get b.

$$\frac{\frac{1}{8}}{\frac{1}{64}ba^{3}}$$

Express \frac{\frac{1}{8}}{\frac{1}{64}ba^{3}} as a single fraction.

$$\frac{1}{8\times \frac{1}{64}ba^{3}}$$

Multiply 8 and \frac{1}{64} to get \frac{1}{8}.

$$\frac{1}{\frac{1}{8}ba^{3}}$$

Differentiate w.r.t. a

$$-\frac{24}{ba^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{8b^{2}\times \frac{1}{64b}}a^{-1-2})$$

Do the arithmetic.

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

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}.

$$-3\times \frac{8}{b}a^{-3-1}$$

Do the arithmetic.

$$\left(-\frac{24}{b}\right)a^{-4}$$