Finite Math Examples

$x^{\frac{5}{9}} - x^{\frac{1}{9}}$

Step 1

Factor $x^{\frac{1}{9}}$ out of $x^{\frac{5}{9}} - x^{\frac{1}{9}}$ .

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Step 1.1

Factor $x^{\frac{1}{9}}$ out of $x^{\frac{5}{9}}$ .

$x^{\frac{1}{9}} x^{\frac{4}{9}} - x^{\frac{1}{9}}$

Step 1.2

Factor $x^{\frac{1}{9}}$ out of $- x^{\frac{1}{9}}$ .

$x^{\frac{1}{9}} x^{\frac{4}{9}} + x^{\frac{1}{9}} \cdot - 1$

Step 1.3

Factor $x^{\frac{1}{9}}$ out of $x^{\frac{1}{9}} x^{\frac{4}{9}} + x^{\frac{1}{9}} \cdot - 1$ .

$x^{\frac{1}{9}} (x^{\frac{4}{9}} - 1)$

Step 2

Rewrite $x^{\frac{4}{9}}$ as ${(x^{\frac{2}{9}})}^{2}$ .

$x^{\frac{1}{9}} ({(x^{\frac{2}{9}})}^{2} - 1)$

Step 3

Rewrite $1$ as $1^{2}$ .

$x^{\frac{1}{9}} ({(x^{\frac{2}{9}})}^{2} - 1^{2})$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{2}{9}}$ and $b = 1$ .

$x^{\frac{1}{9}} ((x^{\frac{2}{9}} + 1) (x^{\frac{2}{9}} - 1))$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Rewrite $x^{\frac{2}{9}}$ as ${(x^{\frac{1}{9}})}^{2}$ .

$x^{\frac{1}{9}} ((x^{\frac{2}{9}} + 1) ({(x^{\frac{1}{9}})}^{2} - 1))$

Step 5.1.2

Rewrite $1$ as $1^{2}$ .

$x^{\frac{1}{9}} ((x^{\frac{2}{9}} + 1) ({(x^{\frac{1}{9}})}^{2} - 1^{2}))$

Step 5.1.3

Factor.

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Step 5.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{1}{9}}$ and $b = 1$ .

$x^{\frac{1}{9}} ((x^{\frac{2}{9}} + 1) ((x^{\frac{1}{9}} + 1) (x^{\frac{1}{9}} - 1)))$

Step 5.1.3.2

Remove unnecessary parentheses.

$x^{\frac{1}{9}} ((x^{\frac{2}{9}} + 1) (x^{\frac{1}{9}} + 1) (x^{\frac{1}{9}} - 1))$

Step 5.2

Remove unnecessary parentheses.

$x^{\frac{1}{9}} (x^{\frac{2}{9}} + 1) (x^{\frac{1}{9}} + 1) (x^{\frac{1}{9}} - 1)$