Basic Math Examples

$a^{101} - a$

Step 1

Factor $a$ out of $a^{101} - a$ .

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Step 1.1

Factor $a$ out of $a^{101}$ .

$a \cdot a^{100} - a$

Step 1.2

Factor $a$ out of $- a$ .

$a \cdot a^{100} + a \cdot - 1$

Step 1.3

Factor $a$ out of $a \cdot a^{100} + a \cdot - 1$ .

$a (a^{100} - 1)$

Step 2

Rewrite $a^{100}$ as ${(a^{50})}^{2}$ .

$a ({(a^{50})}^{2} - 1)$

Step 3

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

$a ({(a^{50})}^{2} - 1^{2})$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a^{50}$ and $b = 1$ .

$a ((a^{50} + 1) (a^{50} - 1))$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Rewrite $a^{50}$ as ${(a^{25})}^{2}$ .

$a ((a^{50} + 1) ({(a^{25})}^{2} - 1))$

Step 5.1.2

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

$a ((a^{50} + 1) ({(a^{25})}^{2} - 1^{2}))$

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 $a = a^{25}$ and $b = 1$ .

$a ((a^{50} + 1) ((a^{25} + 1) (a^{25} - 1)))$

Step 5.1.3.2

Remove unnecessary parentheses.

$a ((a^{50} + 1) (a^{25} + 1) (a^{25} - 1))$

Step 5.2

Remove unnecessary parentheses.

$a (a^{50} + 1) (a^{25} + 1) (a^{25} - 1)$