Solve frac{16a^{2}b^3*2a^-3b^-1}{8a^-2b}

$$\frac { 16 a ^ { 2 } b ^ { 3 } \times 2 a ^ { - 3 } b ^ { - 1 } } { 8 a ^ { - 2 } b }$$

Evaluate

$$4ab$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and -3 to get -1.

$$\frac{16a^{-1}b^{3}\times 2b^{-1}}{8a^{-2}b}$$

To multiply powers of the same base, add their exponents. Add 3 and -1 to get 2.

$$\frac{16a^{-1}b^{2}\times 2}{8a^{-2}b}$$

Cancel out 8b in both numerator and denominator.

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

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

$$2\times 2ba^{1}$$

Multiply 2 and 2 to get 4.

$$4ba^{1}$$

Calculate a to the power of 1 and get a.

$$4ba$$

Differentiate w.r.t. a

$$4b$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{32b^{3}}{ba^{3}\times 8b}a^{2-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{4b}{a^{3}}a^{4})$$

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}.

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

Do the arithmetic.

$$\frac{16b}{a^{3}}a^{3}$$