Finite Math Examples

$| z \cdot 1 \cdot (z \cdot 2) | = | z \cdot 1 | \cdot | z \cdot 2 |$

Step 1

Simplify $| z \cdot 1 \cdot (z \cdot 2) |$ .

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Step 1.1

Rewrite using the commutative property of multiplication.

$| 2 z \cdot z | = | z \cdot 1 | \cdot | z \cdot 2 |$

Step 1.2

Multiply $z$ by $z$ by adding the exponents.

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Step 1.2.1

Move $z$ .

$| 2 (z \cdot z) | = | z \cdot 1 | \cdot | z \cdot 2 |$

Step 1.2.2

Multiply $z$ by $z$ .

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

Step 1.3

Remove non-negative terms from the absolute value.

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

Step 2

Simplify $| z \cdot 1 | \cdot | z \cdot 2 |$ .

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Step 2.1

Multiply $z$ by $1$ .

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

Step 2.2

Move $2$ to the left of $z$ .

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

Step 2.3

Remove non-negative terms from the absolute value.

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

Multiply $2 | z | | z |$ .

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Step 2.5.1

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

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

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

Step 2.5.5

Add $1$ and $1$ .

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

Step 2.6

Remove non-negative terms from the absolute value.

$2 z^{2} = 2 z^{2}$

Step 3

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

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Step 3.1

Subtract $2 z^{2}$ from both sides of the equation.

$2 z^{2} - 2 z^{2} = 0$

Step 3.2

Subtract $2 z^{2}$ from $2 z^{2}$ .

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true for any value of $z$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$