Finite Math Examples

$| z \cdot z^{2} | = | z | \cdot | z^{2} |$

Step 1

Rewrite the equation as $| z | \cdot | z^{2} | = | z \cdot z^{2} |$ .

$| z | \cdot | z^{2} | = | z \cdot z^{2} |$

Step 2

Divide each term in $| z | \cdot | z^{2} | = | z \cdot z^{2} |$ by $| z |$ and simplify.

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Step 2.1

Divide each term in $| z | \cdot | z^{2} | = | z \cdot z^{2} |$ by $| z |$ .

$\frac{| z | \cdot | z^{2} |}{| z |} = \frac{| z \cdot z^{2} |}{| z |}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $| z |$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{| z | \cdot | z^{2} |}{| z |} = \frac{| z \cdot z^{2} |}{| z |}$

Step 2.2.1.2

Divide $| z^{2} |$ by $1$ .

$| z^{2} | = \frac{| z \cdot z^{2} |}{| z |}$

Step 3

Solve for $| z |$ .

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Step 3.1

Rewrite the equation as $\frac{| z \cdot z^{2} |}{| z |} = | z^{2} |$ .

$\frac{| z \cdot z^{2} |}{| z |} = | z^{2} |$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$| z |, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$| z |$

Step 3.3

Multiply each term in $\frac{| z \cdot z^{2} |}{| z |} = | z^{2} |$ by $| z |$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{| z \cdot z^{2} |}{| z |} = | z^{2} |$ by $| z |$ .

$\frac{| z \cdot z^{2} |}{| z |} | z | = | z^{2} | | z |$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $| z |$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{| z \cdot z^{2} |}{| z |} | z | = | z^{2} | | z |$

Step 3.3.2.1.2

Rewrite the expression.

$| z \cdot z^{2} | = | z^{2} | | z |$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $| z^{2} | | z | = | z \cdot z^{2} |$ .

$| z^{2} | | z | = | z \cdot z^{2} |$

Step 3.4.2

Divide each term in $| z^{2} | | z | = | z \cdot z^{2} |$ by $| z^{2} |$ and simplify.

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Step 3.4.2.1

Divide each term in $| z^{2} | | z | = | z \cdot z^{2} |$ by $| z^{2} |$ .

$\frac{| z^{2} | | z |}{| z^{2} |} = \frac{| z \cdot z^{2} |}{| z^{2} |}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $| z^{2} |$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{| z^{2} | | z |}{| z^{2} |} = \frac{| z \cdot z^{2} |}{| z^{2} |}$

Step 3.4.2.2.1.2

Divide $| z |$ by $1$ .

$| z | = \frac{| z \cdot z^{2} |}{| z^{2} |}$

Step 4

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$z = \pm (\frac{| z \cdot z^{2} |}{| z^{2} |})$

Step 5

The result consists of both the positive and negative portions of the $\pm$ .

$z = \frac{| z \cdot z^{2} |}{| z^{2} |}$

$z = - (\frac{| z \cdot z^{2} |}{| z^{2} |})$

Step 6

Solve $z = \frac{| z \cdot z^{2} |}{| z^{2} |}$ for $z$ .

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Step 6.1

Solve for $| z^{2} |$ .

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Step 6.1.1

Rewrite the equation as $\frac{| z \cdot z^{2} |}{| z^{2} |} = z$ .

$\frac{| z \cdot z^{2} |}{| z^{2} |} = z$

Step 6.1.2

Find the LCD of the terms in the equation.

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Step 6.1.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$| z^{2} |, 1$

Step 6.1.2.2

The LCM of one and any expression is the expression.

$| z^{2} |$

Step 6.1.3

Multiply each term in $\frac{| z \cdot z^{2} |}{| z^{2} |} = z$ by $| z^{2} |$ to eliminate the fractions.

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Step 6.1.3.1

Multiply each term in $\frac{| z \cdot z^{2} |}{| z^{2} |} = z$ by $| z^{2} |$ .

$\frac{| z \cdot z^{2} |}{| z^{2} |} | z^{2} | = z | z^{2} |$

Step 6.1.3.2

Simplify the left side.

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Step 6.1.3.2.1

Cancel the common factor of $| z^{2} |$ .

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Step 6.1.3.2.1.1

Cancel the common factor.

$\frac{| z \cdot z^{2} |}{| z^{2} |} | z^{2} | = z | z^{2} |$

Step 6.1.3.2.1.2

Rewrite the expression.

$| z \cdot z^{2} | = z | z^{2} |$

Step 6.1.4

Solve the equation.

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Step 6.1.4.1

Rewrite the equation as $z | z^{2} | = | z \cdot z^{2} |$ .

$z | z^{2} | = | z \cdot z^{2} |$

Step 6.1.4.2

Divide each term in $z | z^{2} | = | z \cdot z^{2} |$ by $z$ and simplify.

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Step 6.1.4.2.1

Divide each term in $z | z^{2} | = | z \cdot z^{2} |$ by $z$ .

$\frac{z | z^{2} |}{z} = \frac{| z \cdot z^{2} |}{z}$

Step 6.1.4.2.2

Simplify the left side.

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Step 6.1.4.2.2.1

Cancel the common factor of $z$ .

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Step 6.1.4.2.2.1.1

Cancel the common factor.

$\frac{z | z^{2} |}{z} = \frac{| z \cdot z^{2} |}{z}$

Step 6.1.4.2.2.1.2

Divide $| z^{2} |$ by $1$ .

$| z^{2} | = \frac{| z \cdot z^{2} |}{z}$

Step 6.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$z^{2} = \pm (\frac{| z \cdot z^{2} |}{z})$

Step 6.3

The result consists of both the positive and negative portions of the $\pm$ .

$z^{2} = \frac{| z \cdot z^{2} |}{z}$

$z^{2} = - (\frac{| z \cdot z^{2} |}{z})$

Step 6.4

Solve $z^{2} = \frac{| z \cdot z^{2} |}{z}$ for $z$ .

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Step 6.4.1

Solve for $| z \cdot z^{2} |$ .

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Step 6.4.1.1

Rewrite the equation as $\frac{| z \cdot z^{2} |}{z} = z^{2}$ .

$\frac{| z \cdot z^{2} |}{z} = z^{2}$

Step 6.4.1.2

Multiply both sides by $z$ .

$\frac{| z \cdot z^{2} |}{z} z = z^{2} z$

Step 6.4.1.3

Simplify.

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Step 6.4.1.3.1

Simplify the left side.

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Step 6.4.1.3.1.1

Cancel the common factor of $z$ .

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Step 6.4.1.3.1.1.1

Cancel the common factor.

$\frac{| z \cdot z^{2} |}{z} z = z^{2} z$

Step 6.4.1.3.1.1.2

Rewrite the expression.

$| z \cdot z^{2} | = z^{2} z$

Step 6.4.1.3.2

Simplify the right side.

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Step 6.4.1.3.2.1

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 6.4.1.3.2.1.1

Multiply $z^{2}$ by $z$ .

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Step 6.4.1.3.2.1.1.1

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

$| z \cdot z^{2} | = z^{2} z^{1}$

Step 6.4.1.3.2.1.1.2

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

$| z \cdot z^{2} | = z^{2 + 1}$

Step 6.4.1.3.2.1.2

Add $2$ and $1$ .

$| z \cdot z^{2} | = z^{3}$

Step 6.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$z \cdot z^{2} = \pm (z^{3})$

Step 6.4.3

The result consists of both the positive and negative portions of the $\pm$ .

$z \cdot z^{2} = z^{3}$

$z \cdot z^{2} = - (z^{3})$

Step 6.4.4

Solve $z \cdot z^{2} = z^{3}$ for $z$ .

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Step 6.4.4.1

Multiply $z$ by $z^{2}$ by adding the exponents.

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Step 6.4.4.1.1

Multiply $z$ by $z^{2}$ .

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Step 6.4.4.1.1.1

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

$z^{1} \cdot z^{2} = z^{3}$

Step 6.4.4.1.1.2

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

$z^{1 + 2} = z^{3}$

Step 6.4.4.1.2

Add $1$ and $2$ .

$z^{3} = z^{3}$

Step 6.4.4.2

Move all terms containing $z$ to the left side of the equation.

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Step 6.4.4.2.1

Subtract $z^{3}$ from both sides of the equation.

$z^{3} - z^{3} = 0$

Step 6.4.4.2.2

Subtract $z^{3}$ from $z^{3}$ .

$0 = 0$

Step 6.4.4.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 6.4.5

Solve $z \cdot z^{2} = - (z^{3})$ for $z$ .

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Step 6.4.5.1

Multiply $z$ by $z^{2}$ by adding the exponents.

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Step 6.4.5.1.1

Multiply $z$ by $z^{2}$ .

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Step 6.4.5.1.1.1

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

$z^{1} \cdot z^{2} = - z^{3}$

Step 6.4.5.1.1.2

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

$z^{1 + 2} = - z^{3}$

Step 6.4.5.1.2

Add $1$ and $2$ .

$z^{3} = - z^{3}$

Step 6.4.5.2

Move all terms containing $z$ to the left side of the equation.

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Step 6.4.5.2.1

Add $z^{3}$ to both sides of the equation.

$z^{3} + z^{3} = 0$

Step 6.4.5.2.2

Add $z^{3}$ and $z^{3}$ .

$2 z^{3} = 0$

Step 6.4.5.3

Divide each term in $2 z^{3} = 0$ by $2$ and simplify.

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Step 6.4.5.3.1

Divide each term in $2 z^{3} = 0$ by $2$ .

$\frac{2 z^{3}}{2} = \frac{0}{2}$

Step 6.4.5.3.2

Simplify the left side.

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Step 6.4.5.3.2.1

Cancel the common factor of $2$ .

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Step 6.4.5.3.2.1.1

Cancel the common factor.

$\frac{2 z^{3}}{2} = \frac{0}{2}$

Step 6.4.5.3.2.1.2

Divide $z^{3}$ by $1$ .

$z^{3} = \frac{0}{2}$

Step 6.4.5.3.3

Simplify the right side.

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Step 6.4.5.3.3.1

Divide $0$ by $2$ .

$z^{3} = 0$

Step 6.4.5.4

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

$z = \sqrt[3]{0}$

Step 6.4.5.5

Simplify $\sqrt[3]{0}$ .

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Step 6.4.5.5.1

Rewrite $0$ as $0^{3}$ .

$z = \sqrt[3]{0^{3}}$

Step 6.4.5.5.2

Pull terms out from under the radical, assuming real numbers.

$z = 0$

Step 6.4.6

Consolidate the solutions.

$z = 0$