Solve frac{12a^{14}b^{8}}{(3a^4b^3)*(2a^2b^3)}

$$\frac { 12 a ^ { 14 } b ^ { 8 } } { ( 3 a ^ { 4 } b ^ { 3 } ) \times ( 2 a ^ { 2 } b ^ { 3 } ) }$$

Evaluate

$$2b^{2}a^{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{12a^{14}b^{8}}{3a^{6}b^{3}\times 2b^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{12a^{14}b^{8}}{3a^{6}b^{6}\times 2}$$

Cancel out 2\times 3a^{6}b^{6} in both numerator and denominator.

$$2b^{2}a^{8}$$

Differentiate w.r.t. a

$$16b^{2}a^{7}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{12b^{8}}{6a^{2}b^{6}}a^{14-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{2b^{2}}{a^{2}}a^{10})$$

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}.

$$10\times \frac{2b^{2}}{a^{2}}a^{10-1}$$

Do the arithmetic.

$$\frac{20b^{2}}{a^{2}}a^{9}$$