Pre-Algebra Examples

$x^{2} \sqrt{x^{4} - 1}$

Step 1

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

$x^{2} \sqrt{{(x^{2})}^{2} - 1}$

Step 2

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

$x^{2} \sqrt{{(x^{2})}^{2} - 1^{2}}$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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