Algebra Examples

$- x^{2} z (2 z^{2} + 4 x z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$- x^{2} z (2 z^{2}) - x^{2} z (4 x z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.2

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

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Step 1.2.1

Move $z^{2}$ .

$- x^{2} (z^{2} z) \cdot 2 - x^{2} z (4 x z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.2.2

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

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Step 1.2.2.1

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

$- x^{2} (z^{2} z^{1}) \cdot 2 - x^{2} z (4 x z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.2.2.2

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

$- x^{2} z^{2 + 1} \cdot 2 - x^{2} z (4 x z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.2.3

Add $2$ and $1$ .

$- x^{2} z^{3} \cdot 2 - x^{2} z (4 x z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.3

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

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Step 1.3.1

Move $x$ .

$- x^{2} z^{3} \cdot 2 - (x \cdot x^{2}) z (4 z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.3.2

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

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Step 1.3.2.1

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

$- x^{2} z^{3} \cdot 2 - (x^{1} x^{2}) z (4 z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.3.2.2

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

$- x^{2} z^{3} \cdot 2 - x^{1 + 2} z (4 z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.3.3

Add $1$ and $2$ .

$- x^{2} z^{3} \cdot 2 - x^{3} z (4 z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $2$ by $- 1$ .

$- 2 x^{2} z^{3} - x^{3} z (4 z^{3}) + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.4.2

Multiply $z$ by $z^{3}$ by adding the exponents.

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Step 1.4.2.1

Move $z^{3}$ .

$- 2 x^{2} z^{3} - x^{3} (z^{3} z) \cdot 4 + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.4.2.2

Multiply $z^{3}$ by $z$ .

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Step 1.4.2.2.1

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

$- 2 x^{2} z^{3} - x^{3} (z^{3} z^{1}) \cdot 4 + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.4.2.2.2

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

$- 2 x^{2} z^{3} - x^{3} z^{3 + 1} \cdot 4 + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.4.2.3

Add $3$ and $1$ .

$- 2 x^{2} z^{3} - x^{3} z^{4} \cdot 4 + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.4.3

Multiply $4$ by $- 1$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x z^{2} (x z + 5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.5

Apply the distributive property.

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x z^{2} (x z) + x z^{2} (5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.6

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

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Step 1.6.1

Move $x$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x \cdot x z^{2} z + x z^{2} (5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.6.2

Multiply $x$ by $x$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{2} z + x z^{2} (5 x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.7

Rewrite using the commutative property of multiplication.

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{2} z + 5 x z^{2} (x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8

Simplify each term.

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Step 1.8.1

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

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Step 1.8.1.1

Move $z$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} (z \cdot z^{2}) + 5 x z^{2} (x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} (z^{1} z^{2}) + 5 x z^{2} (x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.1.2.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{1 + 2} + 5 x z^{2} (x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.1.3

Add $1$ and $2$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x z^{2} (x^{3} z) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.2

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

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Step 1.8.2.1

Move $x^{3}$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 (x^{3} x) z^{2} z + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.2.2

Multiply $x^{3}$ by $x$ .

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Step 1.8.2.2.1

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 (x^{3} x^{1}) z^{2} z + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.2.2.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{3 + 1} z^{2} z + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.2.3

Add $3$ and $1$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{2} z + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.3

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

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Step 1.8.3.1

Move $z$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} (z \cdot z^{2}) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.3.2

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

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Step 1.8.3.2.1

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} (z^{1} z^{2}) + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.3.2.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{1 + 2} + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.8.3.3

Add $1$ and $2$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + x^{2} z^{3} (3 x^{2} z + 4 x z)$

Step 1.9

Apply the distributive property.

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + x^{2} z^{3} (3 x^{2} z) + x^{2} z^{3} (4 x z)$

Step 1.10

Rewrite using the commutative property of multiplication.

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{2} z^{3} (x^{2} z) + x^{2} z^{3} (4 x z)$

Step 1.11

Rewrite using the commutative property of multiplication.

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{2} z^{3} (x^{2} z) + 4 x^{2} z^{3} (x z)$

Step 1.12

Simplify each term.

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Step 1.12.1

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

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Step 1.12.1.1

Move $x^{2}$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 (x^{2} x^{2}) z^{3} z + 4 x^{2} z^{3} (x z)$

Step 1.12.1.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{2 + 2} z^{3} z + 4 x^{2} z^{3} (x z)$

Step 1.12.1.3

Add $2$ and $2$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{3} z + 4 x^{2} z^{3} (x z)$

Step 1.12.2

Multiply $z^{3}$ by $z$ by adding the exponents.

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Step 1.12.2.1

Move $z$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} (z \cdot z^{3}) + 4 x^{2} z^{3} (x z)$

Step 1.12.2.2

Multiply $z$ by $z^{3}$ .

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Step 1.12.2.2.1

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} (z^{1} z^{3}) + 4 x^{2} z^{3} (x z)$

Step 1.12.2.2.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{1 + 3} + 4 x^{2} z^{3} (x z)$

Step 1.12.2.3

Add $1$ and $3$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{2} z^{3} (x z)$

Step 1.12.3

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

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Step 1.12.3.1

Move $x$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 (x \cdot x^{2}) z^{3} z$

Step 1.12.3.2

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

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Step 1.12.3.2.1

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 (x^{1} x^{2}) z^{3} z$

Step 1.12.3.2.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{1 + 2} z^{3} z$

Step 1.12.3.3

Add $1$ and $2$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{3} z^{3} z$

Step 1.12.4

Multiply $z^{3}$ by $z$ by adding the exponents.

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Step 1.12.4.1

Move $z$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{3} (z \cdot z^{3})$

Step 1.12.4.2

Multiply $z$ by $z^{3}$ .

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Step 1.12.4.2.1

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{3} (z^{1} z^{3})$

Step 1.12.4.2.2

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

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{3} z^{1 + 3}$

Step 1.12.4.3

Add $1$ and $3$ .

$- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{3} z^{4}$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $- 2 x^{2} z^{3} - 4 x^{3} z^{4} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 4 x^{3} z^{4}$ .

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Step 2.1.1

Add $- 4 x^{3} z^{4}$ and $4 x^{3} z^{4}$ .

$- 2 x^{2} z^{3} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4} + 0$

Step 2.1.2

Add $- 2 x^{2} z^{3} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4}$ and $0$ .

$- 2 x^{2} z^{3} + x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4}$

Step 2.2

Add $- 2 x^{2} z^{3}$ and $x^{2} z^{3}$ .

$- x^{2} z^{3} + 5 x^{4} z^{3} + 3 x^{4} z^{4}$