Calculus Examples

$x^{\frac{5}{2}} - 16 x^{\frac{1}{2}}$

Step 1

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{5}{2}} - 16 x^{\frac{1}{2}}$ .

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Step 1.1

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{5}{2}}$ .

$x^{\frac{1}{2}} x^{\frac{4}{2}} - 16 x^{\frac{1}{2}}$

Step 1.2

Factor $x^{\frac{1}{2}}$ out of $- 16 x^{\frac{1}{2}}$ .

$x^{\frac{1}{2}} x^{\frac{4}{2}} + x^{\frac{1}{2}} \cdot - 16$

Step 1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{4}{2}} + x^{\frac{1}{2}} \cdot - 16$ .

$x^{\frac{1}{2}} (x^{\frac{4}{2}} - 16)$

Step 2

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

$x^{\frac{1}{2}} ({(x^{\frac{2}{2}})}^{2} - 16)$

Step 3

Rewrite $16$ as $4^{2}$ .

$x^{\frac{1}{2}} ({(x^{\frac{2}{2}})}^{2} - 4^{2})$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{2}{2}}$ and $b = 4$ .

$x^{\frac{1}{2}} ((x^{\frac{2}{2}} + 4) (x^{\frac{2}{2}} - 4))$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Divide $2$ by $2$ .

$x^{\frac{1}{2}} ((x^{1} + 4) (x^{\frac{2}{2}} - 4))$

Step 5.1.2

Simplify.

$x^{\frac{1}{2}} ((x + 4) (x^{\frac{2}{2}} - 4))$

Step 5.1.3

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

$x^{\frac{1}{2}} ((x + 4) ({(x^{\frac{1}{2}})}^{2} - 4))$

Step 5.1.4

Rewrite $4$ as $2^{2}$ .

$x^{\frac{1}{2}} ((x + 4) ({(x^{\frac{1}{2}})}^{2} - 2^{2}))$

Step 5.1.5

Factor.

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Step 5.1.5.1

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

$x^{\frac{1}{2}} ((x + 4) ((x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 2)))$

Step 5.1.5.2

Remove unnecessary parentheses.

$x^{\frac{1}{2}} ((x + 4) (x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 2))$

Step 5.2

Remove unnecessary parentheses.

$x^{\frac{1}{2}} (x + 4) (x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 2)$