Solve frac{5^{-2}b^{9}c^4}{4^-3b^4c^7}

$$\frac { 5 ^ { - 2 } b ^ { 9 } c ^ { 4 } } { 4 ^ { - 3 } b ^ { 4 } c ^ { 7 } }$$

Evaluate

$$\frac{64b^{5}}{25c^{3}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{5^{-2}b^{9}c^{4}}{4^{-3}b^{4}c^{7}}$$

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

$$\frac{5^{-2}}{4^{-3}}b^{9-4}c^{4-7}$$

Subtract 4 from 9.

$$\frac{5^{-2}}{4^{-3}}b^{5}c^{4-7}$$

Subtract 7 from 4.

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

Divide \frac{1}{25} by \frac{1}{64} by multiplying \frac{1}{25} by the reciprocal of \frac{1}{64}.

$$\frac{64}{25}b^{5}\times \frac{1}{c^{3}}$$

Differentiate w.r.t. b

$$\frac{64b^{4}}{5c^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{c^{4}}{25\times \frac{c^{7}}{64}}b^{9-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{64}{25c^{3}}b^{5})$$

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}.

$$5\times \frac{64}{25c^{3}}b^{5-1}$$

Do the arithmetic.

$$\frac{64}{5c^{3}}b^{4}$$