Finite Math Examples

$f (x) = x^{3} \cdot (6 x^{2}) \cdot (11 x) - 6$

Step 1

Set $x^{3} \cdot (6 x^{2}) \cdot (11 x) - 6$ equal to $0$ .

$x^{3} \cdot (6 x^{2}) \cdot (11 x) - 6 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.1.1.1

Move $x^{2}$ .

$x^{2} x^{3} \cdot 6 \cdot (11 x) - 6 = 0$

Step 2.1.1.2

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

$x^{2 + 3} \cdot 6 \cdot (11 x) - 6 = 0$

Step 2.1.1.3

Add $2$ and $3$ .

$x^{5} \cdot 6 \cdot (11 x) - 6 = 0$

Step 2.1.2

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 2.1.2.1

Move $x$ .

$x \cdot x^{5} \cdot 6 \cdot 11 - 6 = 0$

Step 2.1.2.2

Multiply $x$ by $x^{5}$ .

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Step 2.1.2.2.1

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

$x^{1} x^{5} \cdot 6 \cdot 11 - 6 = 0$

Step 2.1.2.2.2

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

$x^{1 + 5} \cdot 6 \cdot 11 - 6 = 0$

Step 2.1.2.3

Add $1$ and $5$ .

$x^{6} \cdot 6 \cdot 11 - 6 = 0$

Step 2.1.3

Move $6$ to the left of $x^{6}$ .

$6 \cdot x^{6} \cdot 11 - 6 = 0$

Step 2.1.4

Multiply $11$ by $6$ .

$66 x^{6} - 6 = 0$

Step 2.2

Add $6$ to both sides of the equation.

$66 x^{6} = 6$

Step 2.3

Divide each term in $66 x^{6} = 6$ by $66$ and simplify.

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Step 2.3.1

Divide each term in $66 x^{6} = 6$ by $66$ .

$\frac{66 x^{6}}{66} = \frac{6}{66}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $66$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{66 x^{6}}{66} = \frac{6}{66}$

Step 2.3.2.1.2

Divide $x^{6}$ by $1$ .

$x^{6} = \frac{6}{66}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $6$ and $66$ .

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Step 2.3.3.1.1

Factor $6$ out of $6$ .

$x^{6} = \frac{6 (1)}{66}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $6$ out of $66$ .

$x^{6} = \frac{6 \cdot 1}{6 \cdot 11}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x^{6} = \frac{6 \cdot 1}{6 \cdot 11}$

Step 2.3.3.1.2.3

Rewrite the expression.

$x^{6} = \frac{1}{11}$

Step 2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[6]{\frac{1}{11}}$

Step 2.5

Simplify $\pm \sqrt[6]{\frac{1}{11}}$ .

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Step 2.5.1

Rewrite $\sqrt[6]{\frac{1}{11}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{11}}$ .

$x = \pm \frac{\sqrt[6]{1}}{\sqrt[6]{11}}$

Step 2.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[6]{11}}$

Step 2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{1}{\sqrt[6]{11}}$

Step 2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{1}{\sqrt[6]{11}}$

Step 2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{\sqrt[6]{11}}, - \frac{1}{\sqrt[6]{11}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{\sqrt[6]{11}}, - \frac{1}{\sqrt[6]{11}}$

Decimal Form:

$x = 0.67055522 \dots, - 0.67055522 \dots$

Step 4