Solve (216x^-3/512)^-1/3

$$( \frac { 216 x ^ { - 3 } } { 512 } ) ^ { - 1 / 3 }$$

Evaluate

$$\frac{4x}{3}$$

Solution Steps

Divide 216x^{-3} by 512 to get \frac{27}{64}x^{-3}.

$$\left(\frac{27}{64}x^{-3}\right)^{-\frac{1}{3}}$$

Expand \left(\frac{27}{64}x^{-3}\right)^{-\frac{1}{3}}.

$$\left(\frac{27}{64}\right)^{-\frac{1}{3}}\left(x^{-3}\right)^{-\frac{1}{3}}$$

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

$$\left(\frac{27}{64}\right)^{-\frac{1}{3}}x^{1}$$

Calculate \frac{27}{64} to the power of -\frac{1}{3} and get \frac{4}{3}.

$$\frac{4}{3}x^{1}$$

Calculate x to the power of 1 and get x.

$$\frac{4}{3}x$$

Differentiate w.r.t. x

$$\frac{4}{3} = 1\frac{1}{3} = 1.3333333333333333$$

Steps Using Definition of a Derivative

Divide 216x^{-3} by 512 to get \frac{27}{64}x^{-3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{27}{64}x^{-3}\right)^{-\frac{1}{3}})$$

Expand \left(\frac{27}{64}x^{-3}\right)^{-\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{27}{64}\right)^{-\frac{1}{3}}\left(x^{-3}\right)^{-\frac{1}{3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{27}{64}\right)^{-\frac{1}{3}}x^{1})$$

Calculate \frac{27}{64} to the power of -\frac{1}{3} and get \frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x^{1})$$

Calculate x to the power of 1 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x)$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{4}{3}x^{1-1}$$

Subtract 1 from 1.

$$\frac{4}{3}x^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{4}{3}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{4}{3}$$