Solve (frac{16b^{5}c^-3}{25b^-5c^5})^-frac{1{2}}

$$( \frac { 16 b ^ { 5 } c ^ { - 3 } } { 25 b ^ { - 5 } c ^ { 5 } } ) ^ { - \frac { 1 } { 2 } }$$

Evaluate

$$\frac{5c^{4}}{4\left(|b|\right)^{5}}$$

Solution Steps

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

$$\left(\frac{16c^{-3}b^{10}}{25c^{5}}\right)^{-\frac{1}{2}}$$

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

$$\left(\frac{16b^{10}}{25c^{8}}\right)^{-\frac{1}{2}}$$

To raise \frac{16b^{10}}{25c^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(16b^{10}\right)^{-\frac{1}{2}}}{\left(25c^{8}\right)^{-\frac{1}{2}}}$$

Expand \left(16b^{10}\right)^{-\frac{1}{2}}.

$$\frac{16^{-\frac{1}{2}}\left(b^{10}\right)^{-\frac{1}{2}}}{\left(25c^{8}\right)^{-\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and -\frac{1}{2} to get -5.

$$\frac{16^{-\frac{1}{2}}b^{-5}}{\left(25c^{8}\right)^{-\frac{1}{2}}}$$

Calculate 16 to the power of -\frac{1}{2} and get \frac{1}{4}.

$$\frac{\frac{1}{4}b^{-5}}{\left(25c^{8}\right)^{-\frac{1}{2}}}$$

Expand \left(25c^{8}\right)^{-\frac{1}{2}}.

$$\frac{\frac{1}{4}b^{-5}}{25^{-\frac{1}{2}}\left(c^{8}\right)^{-\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -\frac{1}{2} to get -4.

$$\frac{\frac{1}{4}b^{-5}}{25^{-\frac{1}{2}}c^{-4}}$$

Calculate 25 to the power of -\frac{1}{2} and get \frac{1}{5}.

$$\frac{\frac{1}{4}b^{-5}}{\frac{1}{5}c^{-4}}$$

Differentiate w.r.t. c

$$\frac{5c^{3}}{\left(|b|\right)^{5}}$$