Finite Math Examples

${(3 z^{3} - 7 z^{4} + 3)}^{2} z^{2}$

Step 1

Rewrite ${(3 z^{3} - 7 z^{4} + 3)}^{2}$ as $(3 z^{3} - 7 z^{4} + 3) (3 z^{3} - 7 z^{4} + 3)$ .

$(3 z^{3} - 7 z^{4} + 3) (3 z^{3} - 7 z^{4} + 3) z^{2}$

Step 2

Expand $(3 z^{3} - 7 z^{4} + 3) (3 z^{3} - 7 z^{4} + 3)$ by multiplying each term in the first expression by each term in the second expression.

$(3 z^{3} (3 z^{3}) + 3 z^{3} (- 7 z^{4}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$(3 \cdot 3 z^{3} z^{3} + 3 z^{3} (- 7 z^{4}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.2

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

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Step 3.1.2.1

Move $z^{3}$ .

$(3 \cdot 3 (z^{3} z^{3}) + 3 z^{3} (- 7 z^{4}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.2.2

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

$(3 \cdot 3 z^{3 + 3} + 3 z^{3} (- 7 z^{4}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.2.3

Add $3$ and $3$ .

$(3 \cdot 3 z^{6} + 3 z^{3} (- 7 z^{4}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.3

Multiply $3$ by $3$ .

$(9 z^{6} + 3 z^{3} (- 7 z^{4}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$(9 z^{6} + 3 \cdot - 7 z^{3} z^{4} + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.5

Multiply $z^{3}$ by $z^{4}$ by adding the exponents.

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Step 3.1.5.1

Move $z^{4}$ .

$(9 z^{6} + 3 \cdot - 7 (z^{4} z^{3}) + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.5.2

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

$(9 z^{6} + 3 \cdot - 7 z^{4 + 3} + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.5.3

Add $4$ and $3$ .

$(9 z^{6} + 3 \cdot - 7 z^{7} + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.6

Multiply $3$ by $- 7$ .

$(9 z^{6} - 21 z^{7} + 3 z^{3} \cdot 3 - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.7

Multiply $3$ by $3$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 7 z^{4} (3 z^{3}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.8

Rewrite using the commutative property of multiplication.

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 7 \cdot 3 z^{4} z^{3} - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.9

Multiply $z^{4}$ by $z^{3}$ by adding the exponents.

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Step 3.1.9.1

Move $z^{3}$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 7 \cdot 3 (z^{3} z^{4}) - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.9.2

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

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 7 \cdot 3 z^{3 + 4} - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.9.3

Add $3$ and $4$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 7 \cdot 3 z^{7} - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.10

Multiply $- 7$ by $3$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} - 7 z^{4} (- 7 z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.11

Rewrite using the commutative property of multiplication.

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} - 7 \cdot - 7 z^{4} z^{4} - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.12

Multiply $z^{4}$ by $z^{4}$ by adding the exponents.

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Step 3.1.12.1

Move $z^{4}$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} - 7 \cdot - 7 (z^{4} z^{4}) - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.12.2

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

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} - 7 \cdot - 7 z^{4 + 4} - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.12.3

Add $4$ and $4$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} - 7 \cdot - 7 z^{8} - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.13

Multiply $- 7$ by $- 7$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} + 49 z^{8} - 7 z^{4} \cdot 3 + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.14

Multiply $3$ by $- 7$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} + 49 z^{8} - 21 z^{4} + 3 (3 z^{3}) + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.15

Multiply $3$ by $3$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} + 49 z^{8} - 21 z^{4} + 9 z^{3} + 3 (- 7 z^{4}) + 3 \cdot 3) z^{2}$

Step 3.1.16

Multiply $- 7$ by $3$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} + 49 z^{8} - 21 z^{4} + 9 z^{3} - 21 z^{4} + 3 \cdot 3) z^{2}$

Step 3.1.17

Multiply $3$ by $3$ .

$(9 z^{6} - 21 z^{7} + 9 z^{3} - 21 z^{7} + 49 z^{8} - 21 z^{4} + 9 z^{3} - 21 z^{4} + 9) z^{2}$

Step 3.2

Simplify terms.

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Step 3.2.1

Subtract $21 z^{7}$ from $- 21 z^{7}$ .

$(9 z^{6} - 42 z^{7} + 9 z^{3} + 49 z^{8} - 21 z^{4} + 9 z^{3} - 21 z^{4} + 9) z^{2}$

Step 3.2.2

Add $9 z^{3}$ and $9 z^{3}$ .

$(9 z^{6} - 42 z^{7} + 49 z^{8} - 21 z^{4} + 18 z^{3} - 21 z^{4} + 9) z^{2}$

Step 3.2.3

Subtract $21 z^{4}$ from $- 21 z^{4}$ .

$(9 z^{6} - 42 z^{7} + 49 z^{8} - 42 z^{4} + 18 z^{3} + 9) z^{2}$

Step 3.2.4

Apply the distributive property.

$9 z^{6} z^{2} - 42 z^{7} z^{2} + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4

Simplify.

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Step 4.1

Multiply $z^{6}$ by $z^{2}$ by adding the exponents.

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Step 4.1.1

Move $z^{2}$ .

$9 (z^{2} z^{6}) - 42 z^{7} z^{2} + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.1.2

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

$9 z^{2 + 6} - 42 z^{7} z^{2} + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.1.3

Add $2$ and $6$ .

$9 z^{8} - 42 z^{7} z^{2} + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.2

Multiply $z^{7}$ by $z^{2}$ by adding the exponents.

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Step 4.2.1

Move $z^{2}$ .

$9 z^{8} - 42 (z^{2} z^{7}) + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.2.2

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

$9 z^{8} - 42 z^{2 + 7} + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.2.3

Add $2$ and $7$ .

$9 z^{8} - 42 z^{9} + 49 z^{8} z^{2} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.3

Multiply $z^{8}$ by $z^{2}$ by adding the exponents.

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Step 4.3.1

Move $z^{2}$ .

$9 z^{8} - 42 z^{9} + 49 (z^{2} z^{8}) - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.3.2

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

$9 z^{8} - 42 z^{9} + 49 z^{2 + 8} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.3.3

Add $2$ and $8$ .

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 z^{4} z^{2} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.4

Multiply $z^{4}$ by $z^{2}$ by adding the exponents.

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Step 4.4.1

Move $z^{2}$ .

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 (z^{2} z^{4}) + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.4.2

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

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 z^{2 + 4} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.4.3

Add $2$ and $4$ .

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 z^{6} + 18 z^{3} z^{2} + 9 z^{2}$

Step 4.5

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

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Step 4.5.1

Move $z^{2}$ .

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 z^{6} + 18 (z^{2} z^{3}) + 9 z^{2}$

Step 4.5.2

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

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 z^{6} + 18 z^{2 + 3} + 9 z^{2}$

Step 4.5.3

Add $2$ and $3$ .

$9 z^{8} - 42 z^{9} + 49 z^{10} - 42 z^{6} + 18 z^{5} + 9 z^{2}$

Step 5

Move $9 z^{8}$ .

$- 42 z^{9} + 49 z^{10} + 9 z^{8} - 42 z^{6} + 18 z^{5} + 9 z^{2}$

Step 6

Reorder $- 42 z^{9}$ and $49 z^{10}$ .

$49 z^{10} - 42 z^{9} + 9 z^{8} - 42 z^{6} + 18 z^{5} + 9 z^{2}$