Determine if Odd, Even, or Neither seventh root of 2x

$\sqrt[7]{2 x}$

Step 1

Write $\sqrt[7]{2 x}$ as a function.

$f (x) = \sqrt[7]{2 x}$

Step 2

Find $f (- x)$ .

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \sqrt[7]{2 (- x)}$

Multiply $- 1$ by $2$ .

$f (- x) = \sqrt[7]{- 2 x}$

Rewrite $- 2 x$ as ${({(- 1)}^{7})}^{7} \cdot (2 x)$ .

Rewrite $- 2$ as $- 1 (2)$ .

$f (- x) = \sqrt[7]{- 1 \cdot 2 x}$

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$f (- x) = \sqrt[7]{{(- 1)}^{7} \cdot (2 x)}$

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$f (- x) = \sqrt[7]{{({(- 1)}^{7})}^{7} \cdot (2 x)}$

Add parentheses.

$f (- x) = \sqrt[7]{{({(- 1)}^{7})}^{7} \cdot (2 x)}$

Pull terms out from under the radical.

$f (- x) = {(- 1)}^{7} \sqrt[7]{2 x}$

Simplify the expression.

Raise $- 1$ to the power of $7$ .

$f (- x) = - 1 \sqrt[7]{2 x}$

Rewrite $- 1 \sqrt[7]{2 x}$ as $- \sqrt[7]{2 x}$ .

$f (- x) = - \sqrt[7]{2 x}$

Step 3

A function is even if $f (- x) = f (x)$ .

Check if $f (- x) = f (x)$ .

Since $- \sqrt[7]{2 x}$ $\neq$ $\sqrt[7]{2 x}$ , the function is not even.

The function is not even

Step 4

A function is odd if $f (- x) = - f (x)$ .

Multiply $- 1$ by $\sqrt[7]{2 x}$ .

$- f (x) = - \sqrt[7]{2 x}$

Since $- \sqrt[7]{2 x} = - \sqrt[7]{2 x}$ , the function is odd.

The function is odd

Step 5