Solve frac{(9a^{4}b^8)^frac{1{2}}}{(6a^3b)^2}=

$$\frac { ( 9 a ^ { 4 } b ^ { 8 } ) ^ { \frac { 1 } { 2 } } } { ( 6 a ^ { 3 } b ) ^ { 2 } } =$$

Evaluate

$$\frac{b^{2}}{12a^{4}}$$

Solution Steps

Expand \left(9a^{4}b^{8}\right)^{\frac{1}{2}}.

$$\frac{9^{\frac{1}{2}}\left(a^{4}\right)^{\frac{1}{2}}\left(b^{8}\right)^{\frac{1}{2}}}{\left(6a^{3}b\right)^{2}}$$

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

$$\frac{9^{\frac{1}{2}}a^{2}\left(b^{8}\right)^{\frac{1}{2}}}{\left(6a^{3}b\right)^{2}}$$

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

$$\frac{9^{\frac{1}{2}}a^{2}b^{4}}{\left(6a^{3}b\right)^{2}}$$

Calculate 9 to the power of \frac{1}{2} and get 3.

$$\frac{3a^{2}b^{4}}{\left(6a^{3}b\right)^{2}}$$

Expand \left(6a^{3}b\right)^{2}.

$$\frac{3a^{2}b^{4}}{6^{2}\left(a^{3}\right)^{2}b^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{3a^{2}b^{4}}{6^{2}a^{6}b^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{3a^{2}b^{4}}{36a^{6}b^{2}}$$

Cancel out 3a^{2}b^{2} in both numerator and denominator.

$$\frac{b^{2}}{12a^{4}}$$

Differentiate w.r.t. b

$$\frac{b}{6a^{4}}$$