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

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

Evaluate

$$\frac{a^{2}}{256\left(bc\right)^{5}}$$

Solution Steps

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

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

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

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

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

$$\frac{\frac{1}{64}a^{2}}{2^{2}b^{5}c^{5}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{1}{64}a^{2}}{4b^{5}c^{5}}$$

Differentiate w.r.t. a

$$\frac{a}{128\left(bc\right)^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{64cb^{2}\times 4b^{3}c^{4}}a^{1-\left(-1\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{256\left(bc\right)^{5}}a^{2})$$

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}.

$$2\times \frac{1}{256\left(bc\right)^{5}}a^{2-1}$$

Do the arithmetic.

$$\frac{1}{128\left(bc\right)^{5}}a^{1}$$

For any term t, t^{1}=t.

$$\frac{1}{128\left(bc\right)^{5}}a$$