Determine if Odd, Even, or Neither x^8-x^10

$x^{8} - x^{10}$

Step 1

Write $x^{8} - x^{10}$ as a function.

$f (x) = x^{8} - x^{10}$

Step 2

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{8} - {(- x)}^{10}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{8} x^{8} - {(- x)}^{10}$

Raise $- 1$ to the power of $8$ .

$f (- x) = 1 x^{8} - {(- x)}^{10}$

Multiply $x^{8}$ by $1$ .

$f (- x) = x^{8} - {(- x)}^{10}$

Apply the product rule to $- x$ .

$f (- x) = x^{8} - ({(- 1)}^{10} x^{10})$

Multiply $- 1$ by ${(- 1)}^{10}$ by adding the exponents.

Move ${(- 1)}^{10}$ .

$f (- x) = x^{8} + {(- 1)}^{10} \cdot (- 1 x^{10})$

Multiply ${(- 1)}^{10}$ by $- 1$ .

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

$f (- x) = x^{8} + {(- 1)}^{10} \cdot ((- 1) x^{10})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = x^{8} + {(- 1)}^{10 + 1} x^{10}$

Add $10$ and $1$ .

$f (- x) = x^{8} + {(- 1)}^{11} x^{10}$

Raise $- 1$ to the power of $11$ .

$f (- x) = x^{8} - x^{10}$

Step 3

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

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

Since $x^{8} - x^{10} = x^{8} - x^{10}$ , the function is even.

The function is even

Step 4