Algebra Examples

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

Step 1

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

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

Step 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 $b = 1$ .

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

Step 3

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

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 1) (x - 1)}$ as ${((x + 1) (x - 1))}^{\frac{1}{2}}$ .

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

Step 3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

${((x + 1) (x - 1))}^{1}$

Step 3.5

Simplify.

$(x + 1) (x - 1)$

Step 4

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$x (x - 1) + 1 (x - 1)$

Step 4.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 1 (x - 1)$

Step 4.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 5.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1$

Step 5.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x + 1 x + 1 \cdot - 1$

Step 5.1.4

Multiply $x$ by $1$ .

$x^{2} - x + x + 1 \cdot - 1$

Step 5.1.5

Multiply $- 1$ by $1$ .

$x^{2} - x + x - 1$

Step 5.2

Add $- x$ and $x$ .

$x^{2} + 0 - 1$

Step 5.3

Add $x^{2}$ and $0$ .

$x^{2} - 1$