Pre-Algebra Examples

$\frac{p^{2} - 16}{\frac{\frac{8 pt}{p^{2} + 12 p + 32}}{4 pt}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(p^{2} - 16) \frac{4 pt}{\frac{8 pt}{p^{2} + 12 p + 32}}$

Step 2

Factor $p^{2} + 12 p + 32$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $32$ and whose sum is $12$ .

$4, 8$

Step 2.2

Write the factored form using these integers.

$(p^{2} - 16) \frac{4 pt}{\frac{8 pt}{(p + 4) (p + 8)}}$

Step 3

Rewrite the expression.

$(p^{2} - 16) \frac{4 pt}{\frac{8}{(p + 4) (p + 8)} pt}$

Step 4

Cancel $pt$ .

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Step 4.1

Cancel $pt$ .

$(p^{2} - 16) \frac{4 pt}{\frac{8}{(p + 4) (p + 8)} pt}$

Step 4.2

Rewrite the expression.

$(p^{2} - 16) \frac{4}{\frac{8}{(p + 4) (p + 8)}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$(p^{2} - 16) (4 \frac{(p + 4) (p + 8)}{8})$

Step 6

Cancel the common factor of $4$ .

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Step 6.1

Factor $4$ out of $8$ .

$(p^{2} - 16) (4 \frac{(p + 4) (p + 8)}{4 (2)})$

Step 6.2

Cancel the common factor.

$(p^{2} - 16) (4 \frac{(p + 4) (p + 8)}{4 \cdot 2})$

Step 6.3

Rewrite the expression.

$(p^{2} - 16) \frac{(p + 4) (p + 8)}{2}$

Step 7

Apply the distributive property.

$p^{2} \frac{(p + 4) (p + 8)}{2} - 16 \frac{(p + 4) (p + 8)}{2}$

Step 8

Combine $p^{2}$ and $\frac{(p + 4) (p + 8)}{2}$ .

$\frac{p^{2} ((p + 4) (p + 8))}{2} - 16 \frac{(p + 4) (p + 8)}{2}$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Factor $2$ out of $- 16$ .

$\frac{p^{2} ((p + 4) (p + 8))}{2} + 2 (- 8) \frac{(p + 4) (p + 8)}{2}$

Step 9.2

Cancel the common factor.

$\frac{p^{2} ((p + 4) (p + 8))}{2} + 2 \cdot - 8 \frac{(p + 4) (p + 8)}{2}$

Step 9.3

Rewrite the expression.

$\frac{p^{2} ((p + 4) (p + 8))}{2} - 8 ((p + 4) (p + 8))$

Step 10

Simplify each term.

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Step 10.1

Expand $(p + 4) (p + 8)$ using the FOIL Method.

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Step 10.1.1

Apply the distributive property.

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p (p + 8) + 4 (p + 8))$

Step 10.1.2

Apply the distributive property.

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p \cdot p + p \cdot 8 + 4 (p + 8))$

Step 10.1.3

Apply the distributive property.

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p \cdot p + p \cdot 8 + 4 p + 4 \cdot 8)$

Step 10.2

Simplify and combine like terms.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Multiply $p$ by $p$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p^{2} + p \cdot 8 + 4 p + 4 \cdot 8)$

Step 10.2.1.2

Move $8$ to the left of $p$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p^{2} + 8 \cdot p + 4 p + 4 \cdot 8)$

Step 10.2.1.3

Multiply $4$ by $8$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p^{2} + 8 p + 4 p + 32)$

Step 10.2.2

Add $8 p$ and $4 p$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 (p^{2} + 12 p + 32)$

Step 10.3

Apply the distributive property.

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 p^{2} - 8 (12 p) - 8 \cdot 32$

Step 10.4

Simplify.

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Step 10.4.1

Multiply $12$ by $- 8$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 p^{2} - 96 p - 8 \cdot 32$

Step 10.4.2

Multiply $- 8$ by $32$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 p^{2} - 96 p - 256$

Step 11

To write $- 8 p^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} - 8 p^{2} \cdot \frac{2}{2} - 96 p - 256$

Step 12

Combine $- 8 p^{2}$ and $\frac{2}{2}$ .

$\frac{p^{2} (p + 4) (p + 8)}{2} + \frac{- 8 p^{2} \cdot 2}{2} - 96 p - 256$

Step 13

Combine the numerators over the common denominator.

$\frac{p^{2} (p + 4) (p + 8) - 8 p^{2} \cdot 2}{2} - 96 p - 256$

Step 14

Simplify the numerator.

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Step 14.1

Factor $p^{2}$ out of $p^{2} (p + 4) (p + 8) - 8 p^{2} \cdot 2$ .

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Step 14.1.1

Factor $p^{2}$ out of $p^{2} (p + 4) (p + 8)$ .

$\frac{p^{2} ((p + 4) (p + 8)) - 8 p^{2} \cdot 2}{2} - 96 p - 256$

Step 14.1.2

Factor $p^{2}$ out of $- 8 p^{2} \cdot 2$ .

$\frac{p^{2} ((p + 4) (p + 8)) + p^{2} (- 8 \cdot 2)}{2} - 96 p - 256$

Step 14.1.3

Factor $p^{2}$ out of $p^{2} ((p + 4) (p + 8)) + p^{2} (- 8 \cdot 2)$ .

$\frac{p^{2} ((p + 4) (p + 8) - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.2

Expand $(p + 4) (p + 8)$ using the FOIL Method.

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Step 14.2.1

Apply the distributive property.

$\frac{p^{2} (p (p + 8) + 4 (p + 8) - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.2.2

Apply the distributive property.

$\frac{p^{2} (p \cdot p + p \cdot 8 + 4 (p + 8) - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.2.3

Apply the distributive property.

$\frac{p^{2} (p \cdot p + p \cdot 8 + 4 p + 4 \cdot 8 - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.3

Simplify and combine like terms.

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Step 14.3.1

Simplify each term.

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Step 14.3.1.1

Multiply $p$ by $p$ .

$\frac{p^{2} (p^{2} + p \cdot 8 + 4 p + 4 \cdot 8 - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.3.1.2

Move $8$ to the left of $p$ .

$\frac{p^{2} (p^{2} + 8 \cdot p + 4 p + 4 \cdot 8 - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.3.1.3

Multiply $4$ by $8$ .

$\frac{p^{2} (p^{2} + 8 p + 4 p + 32 - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.3.2

Add $8 p$ and $4 p$ .

$\frac{p^{2} (p^{2} + 12 p + 32 - 8 \cdot 2)}{2} - 96 p - 256$

Step 14.4

Multiply $- 8$ by $2$ .

$\frac{p^{2} (p^{2} + 12 p + 32 - 16)}{2} - 96 p - 256$

Step 14.5

Subtract $16$ from $32$ .

$\frac{p^{2} (p^{2} + 12 p + 16)}{2} - 96 p - 256$

Step 15

To write $- 96 p$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{p^{2} (p^{2} + 12 p + 16)}{2} - 96 p \cdot \frac{2}{2} - 256$

Step 16

Combine $- 96 p$ and $\frac{2}{2}$ .

$\frac{p^{2} (p^{2} + 12 p + 16)}{2} + \frac{- 96 p \cdot 2}{2} - 256$

Step 17

Combine the numerators over the common denominator.

$\frac{p^{2} (p^{2} + 12 p + 16) - 96 p \cdot 2}{2} - 256$

Step 18

Simplify the numerator.

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Step 18.1

Factor $p$ out of $p^{2} (p^{2} + 12 p + 16) - 96 p \cdot 2$ .

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Step 18.1.1

Factor $p$ out of $p^{2} (p^{2} + 12 p + 16)$ .

$\frac{p (p (p^{2} + 12 p + 16)) - 96 p \cdot 2}{2} - 256$

Step 18.1.2

Factor $p$ out of $- 96 p \cdot 2$ .

$\frac{p (p (p^{2} + 12 p + 16)) + p (- 96 \cdot 2)}{2} - 256$

Step 18.1.3

Factor $p$ out of $p (p (p^{2} + 12 p + 16)) + p (- 96 \cdot 2)$ .

$\frac{p (p (p^{2} + 12 p + 16) - 96 \cdot 2)}{2} - 256$

Step 18.2

Apply the distributive property.

$\frac{p (p \cdot p^{2} + p (12 p) + p \cdot 16 - 96 \cdot 2)}{2} - 256$

Step 18.3

Simplify.

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Step 18.3.1

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

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Step 18.3.1.1

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

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Step 18.3.1.1.1

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

$\frac{p (p^{1} p^{2} + p (12 p) + p \cdot 16 - 96 \cdot 2)}{2} - 256$

Step 18.3.1.1.2

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

$\frac{p (p^{1 + 2} + p (12 p) + p \cdot 16 - 96 \cdot 2)}{2} - 256$

Step 18.3.1.2

Add $1$ and $2$ .

$\frac{p (p^{3} + p (12 p) + p \cdot 16 - 96 \cdot 2)}{2} - 256$

Step 18.3.2

Rewrite using the commutative property of multiplication.

$\frac{p (p^{3} + 12 p \cdot p + p \cdot 16 - 96 \cdot 2)}{2} - 256$

Step 18.3.3

Move $16$ to the left of $p$ .

$\frac{p (p^{3} + 12 p \cdot p + 16 \cdot p - 96 \cdot 2)}{2} - 256$

Step 18.4

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

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Step 18.4.1

Move $p$ .

$\frac{p (p^{3} + 12 (p \cdot p) + 16 \cdot p - 96 \cdot 2)}{2} - 256$

Step 18.4.2

Multiply $p$ by $p$ .

$\frac{p (p^{3} + 12 p^{2} + 16 \cdot p - 96 \cdot 2)}{2} - 256$

$\frac{p (p^{3} + 12 p^{2} + 16 p - 96 \cdot 2)}{2} - 256$

Step 18.5

Multiply $- 96$ by $2$ .

$\frac{p (p^{3} + 12 p^{2} + 16 p - 192)}{2} - 256$

Step 19

To write $- 256$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{p (p^{3} + 12 p^{2} + 16 p - 192)}{2} - 256 \cdot \frac{2}{2}$

Step 20

Combine $- 256$ and $\frac{2}{2}$ .

$\frac{p (p^{3} + 12 p^{2} + 16 p - 192)}{2} + \frac{- 256 \cdot 2}{2}$

Step 21

Combine the numerators over the common denominator.

$\frac{p (p^{3} + 12 p^{2} + 16 p - 192) - 256 \cdot 2}{2}$

Step 22

Simplify the numerator.

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Step 22.1

Apply the distributive property.

$\frac{p \cdot p^{3} + p (12 p^{2}) + p (16 p) + p \cdot - 192 - 256 \cdot 2}{2}$

Step 22.2

Simplify.

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Step 22.2.1

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

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Step 22.2.1.1

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

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Step 22.2.1.1.1

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

$\frac{p^{1} p^{3} + p (12 p^{2}) + p (16 p) + p \cdot - 192 - 256 \cdot 2}{2}$

Step 22.2.1.1.2

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

$\frac{p^{1 + 3} + p (12 p^{2}) + p (16 p) + p \cdot - 192 - 256 \cdot 2}{2}$

Step 22.2.1.2

Add $1$ and $3$ .

$\frac{p^{4} + p (12 p^{2}) + p (16 p) + p \cdot - 192 - 256 \cdot 2}{2}$

Step 22.2.2

Rewrite using the commutative property of multiplication.

$\frac{p^{4} + 12 p \cdot p^{2} + p (16 p) + p \cdot - 192 - 256 \cdot 2}{2}$

Step 22.2.3

Rewrite using the commutative property of multiplication.

$\frac{p^{4} + 12 p \cdot p^{2} + 16 p \cdot p + p \cdot - 192 - 256 \cdot 2}{2}$

Step 22.2.4

Move $- 192$ to the left of $p$ .

$\frac{p^{4} + 12 p \cdot p^{2} + 16 p \cdot p - 192 \cdot p - 256 \cdot 2}{2}$

Step 22.3

Simplify each term.

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Step 22.3.1

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

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Step 22.3.1.1

Move $p^{2}$ .

$\frac{p^{4} + 12 (p^{2} p) + 16 p \cdot p - 192 \cdot p - 256 \cdot 2}{2}$

Step 22.3.1.2

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

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Step 22.3.1.2.1

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

$\frac{p^{4} + 12 (p^{2} p^{1}) + 16 p \cdot p - 192 \cdot p - 256 \cdot 2}{2}$

Step 22.3.1.2.2

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

$\frac{p^{4} + 12 p^{2 + 1} + 16 p \cdot p - 192 \cdot p - 256 \cdot 2}{2}$

Step 22.3.1.3

Add $2$ and $1$ .

$\frac{p^{4} + 12 p^{3} + 16 p \cdot p - 192 \cdot p - 256 \cdot 2}{2}$

Step 22.3.2

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

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Step 22.3.2.1

Move $p$ .

$\frac{p^{4} + 12 p^{3} + 16 (p \cdot p) - 192 \cdot p - 256 \cdot 2}{2}$

Step 22.3.2.2

Multiply $p$ by $p$ .

$\frac{p^{4} + 12 p^{3} + 16 p^{2} - 192 \cdot p - 256 \cdot 2}{2}$

$\frac{p^{4} + 12 p^{3} + 16 p^{2} - 192 p - 256 \cdot 2}{2}$

Step 22.4

Multiply $- 256$ by $2$ .

$\frac{p^{4} + 12 p^{3} + 16 p^{2} - 192 p - 512}{2}$

Step 23

Split the fraction $\frac{p^{4} + 12 p^{3} + 16 p^{2} - 192 p - 512}{2}$ into two fractions.

$\frac{p^{4} + 12 p^{3} + 16 p^{2} - 192 p}{2} + \frac{- 512}{2}$

Step 24

Split the fraction $\frac{p^{4} + 12 p^{3} + 16 p^{2} - 192 p}{2}$ into two fractions.

$\frac{p^{4} + 12 p^{3} + 16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 25

Split the fraction $\frac{p^{4} + 12 p^{3} + 16 p^{2}}{2}$ into two fractions.

$\frac{p^{4} + 12 p^{3}}{2} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 26

Split the fraction $\frac{p^{4} + 12 p^{3}}{2}$ into two fractions.

$\frac{p^{4}}{2} + \frac{12 p^{3}}{2} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 27

Cancel the common factor of $12$ and $2$ .

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Step 27.1

Factor $2$ out of $12 p^{3}$ .

$\frac{p^{4}}{2} + \frac{2 (6 p^{3})}{2} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 27.2

Cancel the common factors.

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Step 27.2.1

Factor $2$ out of $2$ .

$\frac{p^{4}}{2} + \frac{2 (6 p^{3})}{2 (1)} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 27.2.2

Cancel the common factor.

$\frac{p^{4}}{2} + \frac{2 (6 p^{3})}{2 \cdot 1} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 27.2.3

Rewrite the expression.

$\frac{p^{4}}{2} + \frac{6 p^{3}}{1} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 27.2.4

Divide $6 p^{3}$ by $1$ .

$\frac{p^{4}}{2} + 6 p^{3} + \frac{16 p^{2}}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 28

Cancel the common factor of $16$ and $2$ .

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Step 28.1

Factor $2$ out of $16 p^{2}$ .

$\frac{p^{4}}{2} + 6 p^{3} + \frac{2 (8 p^{2})}{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 28.2

Cancel the common factors.

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Step 28.2.1

Factor $2$ out of $2$ .

$\frac{p^{4}}{2} + 6 p^{3} + \frac{2 (8 p^{2})}{2 (1)} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 28.2.2

Cancel the common factor.

$\frac{p^{4}}{2} + 6 p^{3} + \frac{2 (8 p^{2})}{2 \cdot 1} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 28.2.3

Rewrite the expression.

$\frac{p^{4}}{2} + 6 p^{3} + \frac{8 p^{2}}{1} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 28.2.4

Divide $8 p^{2}$ by $1$ .

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} + \frac{- 192 p}{2} + \frac{- 512}{2}$

Step 29

Cancel the common factor of $- 192$ and $2$ .

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Step 29.1

Factor $2$ out of $- 192 p$ .

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} + \frac{2 (- 96 p)}{2} + \frac{- 512}{2}$

Step 29.2

Cancel the common factors.

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Step 29.2.1

Factor $2$ out of $2$ .

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} + \frac{2 (- 96 p)}{2 (1)} + \frac{- 512}{2}$

Step 29.2.2

Cancel the common factor.

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} + \frac{2 (- 96 p)}{2 \cdot 1} + \frac{- 512}{2}$

Step 29.2.3

Rewrite the expression.

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} + \frac{- 96 p}{1} + \frac{- 512}{2}$

Step 29.2.4

Divide $- 96 p$ by $1$ .

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} - 96 p + \frac{- 512}{2}$

Step 30

Divide $- 512$ by $2$ .

$\frac{p^{4}}{2} + 6 p^{3} + 8 p^{2} - 96 p - 256$