Solve frac{ab^{-8}cd^{7}}{a^{-9}b^{8}c^{0}d^-6*ab^-1c^2d^5}

$$\frac { a b ^ { - 8 } c d ^ { 7 } } { a ^ { - 9 } b ^ { 8 } c ^ { 0 } d ^ { - 6 } \cdot a b ^ { - 1 } c ^ { 2 } d ^ { 5 } }$$

Evaluate

$$\frac{d^{8}a^{9}}{cb^{15}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -9 and 1 to get -8.

$$\frac{ab^{-8}cd^{7}}{a^{-8}b^{8}c^{0}d^{-6}b^{-1}c^{2}d^{5}}$$

To multiply powers of the same base, add their exponents. Add 8 and -1 to get 7.

$$\frac{ab^{-8}cd^{7}}{a^{-8}b^{7}c^{0}d^{-6}c^{2}d^{5}}$$

To multiply powers of the same base, add their exponents. Add 0 and 2 to get 2.

$$\frac{ab^{-8}cd^{7}}{a^{-8}b^{7}c^{2}d^{-6}d^{5}}$$

To multiply powers of the same base, add their exponents. Add -6 and 5 to get -1.

$$\frac{ab^{-8}cd^{7}}{a^{-8}b^{7}c^{2}d^{-1}}$$

Cancel out c in both numerator and denominator.

$$\frac{b^{-8}ad^{7}}{a^{-8}\times \frac{1}{d}cb^{7}}$$

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

$$\frac{b^{-8}d^{8}a^{9}}{cb^{7}}$$

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

$$\frac{d^{8}a^{9}}{cb^{15}}$$

Differentiate w.r.t. a

$$\frac{9\left(ad\right)^{8}}{cb^{15}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{cd^{7}}{b^{8}\times \frac{ac^{0}c^{2}d^{5}b^{8}}{bd^{6}}}a^{1-\left(-9\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{d^{8}}{acb^{15}}a^{10})$$

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}.

$$10\times \frac{d^{8}}{acb^{15}}a^{10-1}$$

Do the arithmetic.

$$\frac{10d^{8}}{acb^{15}}a^{9}$$