Solve [16a^-4div25b^-2]^frac{1{2}}

$$[ 16 a ^ { - 4 } \div 25 b ^ { - 2 } ] ^ { \frac { 1 } { 2 } }$$

Evaluate

$$\frac{4}{5|b|a^{2}}$$

Solution Steps

Express \frac{16a^{-4}}{25}b^{-2} as a single fraction.

$$\left(\frac{16a^{-4}b^{-2}}{25}\right)^{\frac{1}{2}}$$

To raise \frac{16a^{-4}b^{-2}}{25} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(16a^{-4}b^{-2}\right)^{\frac{1}{2}}}{25^{\frac{1}{2}}}$$

Expand \left(16a^{-4}b^{-2}\right)^{\frac{1}{2}}.

$$\frac{16^{\frac{1}{2}}\left(a^{-4}\right)^{\frac{1}{2}}\left(b^{-2}\right)^{\frac{1}{2}}}{25^{\frac{1}{2}}}$$

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

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

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

$$\frac{16^{\frac{1}{2}}a^{-2}b^{-1}}{25^{\frac{1}{2}}}$$

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

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

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

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

Differentiate w.r.t. b

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