Solve (27x^6)^frac{5{3}}

$$( 27 x ^ { 6 } ) ^ { \frac { 5 } { 3 } }$$

Evaluate

$$243x^{10}$$

Solution Steps

Expand \left(27x^{6}\right)^{\frac{5}{3}}.

$$27^{\frac{5}{3}}\left(x^{6}\right)^{\frac{5}{3}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{5}{3} to get 10.

$$27^{\frac{5}{3}}x^{10}$$

Calculate 27 to the power of \frac{5}{3} and get 243.

$$243x^{10}$$

Differentiate w.r.t. x

$$2430x^{9}$$

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).

$$\frac{5}{3}\times \left(27x^{6}\right)^{\frac{5}{3}-1}\frac{\mathrm{d}}{\mathrm{d}x}(27x^{6})$$

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}.

$$\frac{5}{3}\times \left(27x^{6}\right)^{\frac{2}{3}}\times 6\times 27x^{6-1}$$

Simplify.

$$270x^{5}\times \left(27x^{6}\right)^{\frac{2}{3}}$$