Basic Math Examples

${(p^{2} - 12 p + 36)}^{2}$

Step 1

Rewrite ${(p^{2} - 12 p + 36)}^{2}$ as $(p^{2} - 12 p + 36) (p^{2} - 12 p + 36)$ .

$(p^{2} - 12 p + 36) (p^{2} - 12 p + 36)$

Step 2

Expand $(p^{2} - 12 p + 36) (p^{2} - 12 p + 36)$ by multiplying each term in the first expression by each term in the second expression.

$p^{2} p^{2} + p^{2} (- 12 p) + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

$p^{2 + 2} + p^{2} (- 12 p) + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.1.2

Add $2$ and $2$ .

$p^{4} + p^{2} (- 12 p) + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$p^{4} - 12 p^{2} p + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.3

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

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Step 3.1.3.1

Move $p$ .

$p^{4} - 12 (p \cdot p^{2}) + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.3.2

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

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Step 3.1.3.2.1

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

$p^{4} - 12 (p^{1} p^{2}) + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.3.2.2

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

$p^{4} - 12 p^{1 + 2} + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.3.3

Add $1$ and $2$ .

$p^{4} - 12 p^{3} + p^{2} \cdot 36 - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.4

Move $36$ to the left of $p^{2}$ .

$p^{4} - 12 p^{3} + 36 \cdot p^{2} - 12 p \cdot p^{2} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.5

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

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Step 3.1.5.1

Move $p^{2}$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 (p^{2} p) - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.5.2

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

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Step 3.1.5.2.1

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

$p^{4} - 12 p^{3} + 36 p^{2} - 12 (p^{2} p^{1}) - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.5.2.2

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

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{2 + 1} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.5.3

Add $2$ and $1$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} - 12 p (- 12 p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} - 12 \cdot - 12 p \cdot p - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.7

Multiply $p$ by $p$ by adding the exponents.

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Step 3.1.7.1

Move $p$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} - 12 \cdot - 12 (p \cdot p) - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.7.2

Multiply $p$ by $p$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} - 12 \cdot - 12 p^{2} - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.8

Multiply $- 12$ by $- 12$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} + 144 p^{2} - 12 p \cdot 36 + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.9

Multiply $36$ by $- 12$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} + 144 p^{2} - 432 p + 36 p^{2} + 36 (- 12 p) + 36 \cdot 36$

Step 3.1.10

Multiply $- 12$ by $36$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} + 144 p^{2} - 432 p + 36 p^{2} - 432 p + 36 \cdot 36$

Step 3.1.11

Multiply $36$ by $36$ .

$p^{4} - 12 p^{3} + 36 p^{2} - 12 p^{3} + 144 p^{2} - 432 p + 36 p^{2} - 432 p + 1296$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $12 p^{3}$ from $- 12 p^{3}$ .

$p^{4} - 24 p^{3} + 36 p^{2} + 144 p^{2} - 432 p + 36 p^{2} - 432 p + 1296$

Step 3.2.2

Add $36 p^{2}$ and $144 p^{2}$ .

$p^{4} - 24 p^{3} + 180 p^{2} - 432 p + 36 p^{2} - 432 p + 1296$

Step 3.2.3

Add $180 p^{2}$ and $36 p^{2}$ .

$p^{4} - 24 p^{3} + 216 p^{2} - 432 p - 432 p + 1296$

Step 3.2.4

Subtract $432 p$ from $- 432 p$ .

$p^{4} - 24 p^{3} + 216 p^{2} - 864 p + 1296$