Solve {left(81y^16/16x^12right)}^frac{1{2}}

$${ \left( \frac{ 81 { y }^{ 16 } }{ 16 { x }^{ 12 } } \right) }^{ \frac{ 1 }{ 2 } }$$

Evaluate

$$\frac{9y^{8}}{4x^{6}}$$

Solution Steps

To raise \frac{81y^{16}}{16x^{12}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(81y^{16}\right)^{\frac{1}{2}}}{\left(16x^{12}\right)^{\frac{1}{2}}}$$

Expand \left(81y^{16}\right)^{\frac{1}{2}}.

$$\frac{81^{\frac{1}{2}}\left(y^{16}\right)^{\frac{1}{2}}}{\left(16x^{12}\right)^{\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 16 and \frac{1}{2} to get 8.

$$\frac{81^{\frac{1}{2}}y^{8}}{\left(16x^{12}\right)^{\frac{1}{2}}}$$

Calculate 81 to the power of \frac{1}{2} and get 9.

$$\frac{9y^{8}}{\left(16x^{12}\right)^{\frac{1}{2}}}$$

Expand \left(16x^{12}\right)^{\frac{1}{2}}.

$$\frac{9y^{8}}{16^{\frac{1}{2}}\left(x^{12}\right)^{\frac{1}{2}}}$$

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

$$\frac{9y^{8}}{16^{\frac{1}{2}}x^{6}}$$

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

$$\frac{9y^{8}}{4x^{6}}$$

Differentiate w.r.t. x

$$-\frac{27y^{8}}{2x^{7}}$$