Solve {left(a^4b^-6c^2/a^6{b^6{c}^-4}right)}^-frac{1{2}}

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

Evaluate

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

Solution Steps

Cancel out a^{4} in both numerator and denominator.

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

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

$$\left(\frac{b^{-6}c^{6}}{a^{2}b^{6}}\right)^{-\frac{1}{2}}$$

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

$$\left(\frac{c^{6}}{a^{2}b^{12}}\right)^{-\frac{1}{2}}$$

To raise \frac{c^{6}}{a^{2}b^{12}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(c^{6}\right)^{-\frac{1}{2}}}{\left(a^{2}b^{12}\right)^{-\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -\frac{1}{2} to get -3.

$$\frac{c^{-3}}{\left(a^{2}b^{12}\right)^{-\frac{1}{2}}}$$

Expand \left(a^{2}b^{12}\right)^{-\frac{1}{2}}.

$$\frac{c^{-3}}{\left(a^{2}\right)^{-\frac{1}{2}}\left(b^{12}\right)^{-\frac{1}{2}}}$$

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

$$\frac{c^{-3}}{a^{-1}\left(b^{12}\right)^{-\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 12 and -\frac{1}{2} to get -6.

$$\frac{c^{-3}}{a^{-1}b^{-6}}$$

Differentiate w.r.t. a

$$\frac{|a|b^{6}}{a\left(|c|\right)^{3}}$$