Solve (frac{3a^{-2}bc^-3}{24a^5b^-3c})^-1

$$( \frac { 3 a ^ { - 2 } b c ^ { - 3 } } { 24 a ^ { 5 } b ^ { - 3 } c } ) ^ { - 1 }$$

Evaluate

$$8\times \left(\frac{c}{b}\right)^{4}a^{7}$$

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\left(\frac{c^{-3}a^{-2}b}{8b^{-3}ca^{5}}\right)^{-1}$$

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

$$\left(\frac{c^{-3}a^{-2}b^{4}}{8ca^{5}}\right)^{-1}$$

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

$$\left(\frac{b^{4}}{8c^{4}a^{7}}\right)^{-1}$$

To raise \frac{b^{4}}{8c^{4}a^{7}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(b^{4}\right)^{-1}}{\left(8c^{4}a^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$\frac{b^{-4}}{\left(8c^{4}a^{7}\right)^{-1}}$$

Expand \left(8c^{4}a^{7}\right)^{-1}.

$$\frac{b^{-4}}{8^{-1}\left(c^{4}\right)^{-1}\left(a^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$\frac{b^{-4}}{8^{-1}c^{-4}\left(a^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -1 to get -7.

$$\frac{b^{-4}}{8^{-1}c^{-4}a^{-7}}$$

Calculate 8 to the power of -1 and get \frac{1}{8}.

$$\frac{b^{-4}}{\frac{1}{8}c^{-4}a^{-7}}$$

Differentiate w.r.t. c

$$\frac{32c^{3}a^{7}}{b^{4}}$$