Algebra Examples

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

Step 1

Factor $x^{2} - x - 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2

Write the factored form using these integers.

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

Step 2

Combine $\sqrt{x - 2}$ and $\sqrt{(x - 2) (x + 1)}$ into a single radical.

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

Step 3

Reduce the expression $\frac{x - 2}{(x - 2) (x + 1)}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$(x + 1) \sqrt{\frac{1}{x + 1}}$

Step 4

Rewrite $\sqrt{\frac{1}{x + 1}}$ as $\frac{\sqrt{1}}{\sqrt{x + 1}}$ .

$(x + 1) \frac{\sqrt{1}}{\sqrt{x + 1}}$

Step 5

Any root of $1$ is $1$ .

$(x + 1) \frac{1}{\sqrt{x + 1}}$

Step 6

Multiply $\frac{1}{\sqrt{x + 1}}$ by $\frac{\sqrt{x + 1}}{\sqrt{x + 1}}$ .

$(x + 1) (\frac{1}{\sqrt{x + 1}} \cdot \frac{\sqrt{x + 1}}{\sqrt{x + 1}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{x + 1}}$ by $\frac{\sqrt{x + 1}}{\sqrt{x + 1}}$ .

$(x + 1) \frac{\sqrt{x + 1}}{\sqrt{x + 1} \sqrt{x + 1}}$

Step 7.2

Raise $\sqrt{x + 1}$ to the power of $1$ .

$(x + 1) \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{1} \sqrt{x + 1}}$

Step 7.3

Raise $\sqrt{x + 1}$ to the power of $1$ .

$(x + 1) \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1}}$

Step 7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 1) \frac{\sqrt{x + 1}}{{\sqrt{x + 1}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

$(x + 1) \frac{\sqrt{x + 1}}{{(x + 1)}^{1}}$

Step 7.6.5

Simplify.

$(x + 1) \frac{\sqrt{x + 1}}{x + 1}$

Step 8

Cancel the common factor of $x + 1$ .

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Step 8.1

Cancel the common factor.

$(x + 1) \frac{\sqrt{x + 1}}{x + 1}$

Step 8.2

Rewrite the expression.

$\sqrt{x + 1}$