Solve (8x^15)^-frac{1{3}}

$$( 8 x ^ { 15 } ) ^ { - \frac { 1 } { 3 } }$$

Evaluate

$$\frac{1}{2x^{5}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\left(8x^{15}\right)^{-\frac{1}{3}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$8^{-\frac{1}{3}}\left(x^{15}\right)^{-\frac{1}{3}}$$

Raise 8 to the power -\frac{1}{3}.

$$\frac{1}{2}\left(x^{15}\right)^{-\frac{1}{3}}$$

To raise a power to another power, multiply the exponents.

$$\frac{1}{2}x^{15\left(-\frac{1}{3}\right)}$$

Multiply 15 times -\frac{1}{3}.

$$\frac{1}{2}\times \frac{1}{x^{5}}$$

Differentiate w.r.t. x

$$-\frac{5}{2x^{6}}$$

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).

$$-\frac{1}{3}\times \left(8x^{15}\right)^{-\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}x}(8x^{15})$$

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}.

$$-\frac{1}{3}\times \left(8x^{15}\right)^{-\frac{4}{3}}\times 15\times 8x^{15-1}$$

Simplify.

$$-40x^{14}\times \left(8x^{15}\right)^{-\frac{4}{3}}$$