Solve frac{e^6{left(8k^3right)}^frac{1{3}}}{{left(e^2k^4right)}^frac{1{2}}}

$$\frac{ { e }^{ 6 } { \left(8 { k }^{ 3 } \right) }^{ \frac{ 1 }{ 3 } } }{ { \left( { e }^{ 2 } { k }^{ 4 } \right) }^{ \frac{ 1 }{ 2 } } }$$

Evaluate

$$\frac{2e^{5}}{k}$$

Solution Steps

Expand \left(8k^{3}\right)^{\frac{1}{3}}.

$$\frac{e^{6}\times 8^{\frac{1}{3}}\left(k^{3}\right)^{\frac{1}{3}}}{\left(e^{2}k^{4}\right)^{\frac{1}{2}}}$$

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

$$\frac{e^{6}\times 8^{\frac{1}{3}}k^{1}}{\left(e^{2}k^{4}\right)^{\frac{1}{2}}}$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{e^{6}\times 2k^{1}}{\left(e^{2}k^{4}\right)^{\frac{1}{2}}}$$

Calculate k to the power of 1 and get k.

$$\frac{e^{6}\times 2k}{\left(e^{2}k^{4}\right)^{\frac{1}{2}}}$$

Expand \left(e^{2}k^{4}\right)^{\frac{1}{2}}.

$$\frac{e^{6}\times 2k}{\left(e^{2}\right)^{\frac{1}{2}}\left(k^{4}\right)^{\frac{1}{2}}}$$

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

$$\frac{e^{6}\times 2k}{e^{1}\left(k^{4}\right)^{\frac{1}{2}}}$$

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

$$\frac{e^{6}\times 2k}{e^{1}k^{2}}$$

Calculate e to the power of 1 and get e.

$$\frac{e^{6}\times 2k}{ek^{2}}$$

Cancel out ek in both numerator and denominator.

$$\frac{2e^{5}}{k}$$

Differentiate w.r.t. k

$$-\frac{2e^{5}}{k^{2}}$$