Solve (8p^6)^frac{1{3}}

$$( 8 p ^ { 6 } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$2p^{2}$$

Solution Steps

Expand \left(8p^{6}\right)^{\frac{1}{3}}.

$$8^{\frac{1}{3}}\left(p^{6}\right)^{\frac{1}{3}}$$

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

$$8^{\frac{1}{3}}p^{2}$$

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

$$2p^{2}$$

Differentiate w.r.t. p

$$4p$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\frac{1}{3}\times \left(8p^{6}\right)^{\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}p}(8p^{6})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{3}\times \left(8p^{6}\right)^{-\frac{2}{3}}\times 6\times 8p^{6-1}$$

Simplify.

$$16p^{5}\times \left(8p^{6}\right)^{-\frac{2}{3}}$$