Finite Math Examples

$\frac{- x^{3} + x}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Factor $x$ out of $- x^{3} + x$ .

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Step 1.1.1.1

Factor $x$ out of $- x^{3}$ .

$\frac{x (- x^{2}) + x}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.1.1.2

Raise $x$ to the power of $1$ .

$\frac{x (- x^{2}) + x^{1}}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.1.1.3

Factor $x$ out of $x^{1}$ .

$\frac{x (- x^{2}) + x \cdot 1}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.1.1.4

Factor $x$ out of $x (- x^{2}) + x \cdot 1$ .

$\frac{x (- x^{2} + 1)}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.1.2

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

$\frac{x (- x^{2} + 1^{2})}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.1.3

Reorder $- x^{2}$ and $1^{2}$ .

$\frac{x (1^{2} - x^{2})}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.1.4

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

$\frac{x (1 + x) (1 - x)}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.2

Multiply $\frac{x (1 + x) (1 - x)}{\sqrt{13 - x^{2}}}$ by $\frac{\sqrt{13 - x^{2}}}{\sqrt{13 - x^{2}}}$ .

$\frac{x (1 + x) (1 - x)}{\sqrt{13 - x^{2}}} \cdot \frac{\sqrt{13 - x^{2}}}{\sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{x (1 + x) (1 - x)}{\sqrt{13 - x^{2}}}$ by $\frac{\sqrt{13 - x^{2}}}{\sqrt{13 - x^{2}}}$ .

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{\sqrt{13 - x^{2}} \sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.2

Raise $\sqrt{13 - x^{2}}$ to the power of $1$ .

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{\sqrt{13 - x^{2}}}^{1} \sqrt{13 - x^{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.3

Raise $\sqrt{13 - x^{2}}$ to the power of $1$ .

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{\sqrt{13 - x^{2}}}^{1} {\sqrt{13 - x^{2}}}^{1}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.4

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

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{\sqrt{13 - x^{2}}}^{1 + 1}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{\sqrt{13 - x^{2}}}^{2}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.6

Rewrite ${\sqrt{13 - x^{2}}}^{2}$ as $13 - x^{2}$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13 - x^{2}}$ as ${(13 - x^{2})}^{\frac{1}{2}}$ .

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{({(13 - x^{2})}^{\frac{1}{2}})}^{2}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.6.2

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

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{(13 - x^{2})}^{\frac{1}{2} \cdot 2}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.6.3

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

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{(13 - x^{2})}^{\frac{2}{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{(13 - x^{2})}^{\frac{2}{2}}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{{(13 - x^{2})}^{1}} + 12 x \sqrt{13 - x^{2}}$

Step 1.3.6.5

Simplify.

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{13 - x^{2}} + 12 x \sqrt{13 - x^{2}}$

Step 2

To write $12 x \sqrt{13 - x^{2}}$ as a fraction with a common denominator, multiply by $\frac{13 - x^{2}}{13 - x^{2}}$ .

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}}}{13 - x^{2}} + \frac{12 x \sqrt{13 - x^{2}} (13 - x^{2})}{13 - x^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{x (1 + x) (1 - x) \sqrt{13 - x^{2}} + 12 x \sqrt{13 - x^{2}} (13 - x^{2})}{13 - x^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x \sqrt{13 - x^{2}}$ out of $x (1 + x) (1 - x) \sqrt{13 - x^{2}} + 12 x \sqrt{13 - x^{2}} (13 - x^{2})$ .

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Step 4.1.1

Factor $x \sqrt{13 - x^{2}}$ out of $x (1 + x) (1 - x) \sqrt{13 - x^{2}}$ .

$\frac{x \sqrt{13 - x^{2}} ((1 + x) (1 - x)) + 12 x \sqrt{13 - x^{2}} (13 - x^{2})}{13 - x^{2}}$

Step 4.1.2

Factor $x \sqrt{13 - x^{2}}$ out of $12 x \sqrt{13 - x^{2}} (13 - x^{2})$ .

$\frac{x \sqrt{13 - x^{2}} ((1 + x) (1 - x)) + x \sqrt{13 - x^{2}} (12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.1.3

Factor $x \sqrt{13 - x^{2}}$ out of $x \sqrt{13 - x^{2}} ((1 + x) (1 - x)) + x \sqrt{13 - x^{2}} (12 (13 - x^{2}))$ .

$\frac{x \sqrt{13 - x^{2}} ((1 + x) (1 - x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

$\frac{x \sqrt{13 - x^{2}} (1 (1 - x) + x (1 - x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.2.2

Apply the distributive property.

$\frac{x \sqrt{13 - x^{2}} (1 \cdot 1 + 1 (- x) + x (1 - x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.2.3

Apply the distributive property.

$\frac{x \sqrt{13 - x^{2}} (1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $1$ by $1$ .

$\frac{x \sqrt{13 - x^{2}} (1 + 1 (- x) + x \cdot 1 + x (- x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.1.2

Multiply $- x$ by $1$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x + x \cdot 1 + x (- x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.1.3

Multiply $x$ by $1$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x + x + x (- x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{x \sqrt{13 - x^{2}} (1 - x + x - x \cdot x + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.5.1

Move $x$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x + x - (x \cdot x) + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.1.5.2

Multiply $x$ by $x$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x + x - x^{2} + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.2

Add $- x$ and $x$ .

$\frac{x \sqrt{13 - x^{2}} (1 + 0 - x^{2} + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.3.3

Add $1$ and $0$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x^{2} + 12 (13 - x^{2}))}{13 - x^{2}}$

Step 4.4

Apply the distributive property.

$\frac{x \sqrt{13 - x^{2}} (1 - x^{2} + 12 \cdot 13 + 12 (- x^{2}))}{13 - x^{2}}$

Step 4.5

Multiply $12$ by $13$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x^{2} + 156 + 12 (- x^{2}))}{13 - x^{2}}$

Step 4.6

Multiply $- 1$ by $12$ .

$\frac{x \sqrt{13 - x^{2}} (1 - x^{2} + 156 - 12 x^{2})}{13 - x^{2}}$

Step 4.7

Add $1$ and $156$ .

$\frac{x \sqrt{13 - x^{2}} (- x^{2} + 157 - 12 x^{2})}{13 - x^{2}}$

Step 4.8

Subtract $12 x^{2}$ from $- x^{2}$ .

$\frac{x \sqrt{13 - x^{2}} (- 13 x^{2} + 157)}{13 - x^{2}}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 13 x^{2}$ .

$\frac{x \sqrt{13 - x^{2}} (- (13 x^{2}) + 157)}{13 - x^{2}}$

Step 5.2

Rewrite $157$ as $- 1 (- 157)$ .

$\frac{x \sqrt{13 - x^{2}} (- (13 x^{2}) - 1 (- 157))}{13 - x^{2}}$

Step 5.3

Factor $- 1$ out of $- (13 x^{2}) - 1 (- 157)$ .

$\frac{x \sqrt{13 - x^{2}} (- (13 x^{2} - 157))}{13 - x^{2}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (13 x^{2} - 157)$ as $- 1 (13 x^{2} - 157)$ .

$\frac{x \sqrt{13 - x^{2}} (- 1 (13 x^{2} - 157))}{13 - x^{2}}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{(x \sqrt{13 - x^{2}}) (13 x^{2} - 157)}{13 - x^{2}}$

Step 5.4.3

Reorder factors in $- \frac{(x \sqrt{13 - x^{2}}) (13 x^{2} - 157)}{13 - x^{2}}$ .

$- \frac{x \sqrt{13 - x^{2}} (13 x^{2} - 157)}{13 - x^{2}}$