Basic Math Examples

$(5 p^{2} + 3 p) (p^{3} - p^{2} + 5)$

Step 1

Expand $(5 p^{2} + 3 p) (p^{3} - p^{2} + 5)$ by multiplying each term in the first expression by each term in the second expression.

$5 p^{2} p^{3} + 5 p^{2} (- p^{2}) + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Move $p^{3}$ .

$5 (p^{3} p^{2}) + 5 p^{2} (- p^{2}) + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.1.2

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

$5 p^{3 + 2} + 5 p^{2} (- p^{2}) + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.1.3

Add $3$ and $2$ .

$5 p^{5} + 5 p^{2} (- p^{2}) + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$5 p^{5} + 5 \cdot - 1 p^{2} p^{2} + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.3

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

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Step 2.1.3.1

Move $p^{2}$ .

$5 p^{5} + 5 \cdot - 1 (p^{2} p^{2}) + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.3.2

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

$5 p^{5} + 5 \cdot - 1 p^{2 + 2} + 5 p^{2} \cdot 5 + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.3.3

Add $2$ and $2$ .

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

Step 2.1.4

Multiply $5$ by $- 1$ .

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

Step 2.1.5

Multiply $5$ by $5$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p \cdot p^{3} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.6

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

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Step 2.1.6.1

Move $p^{3}$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 (p^{3} p) + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.6.2

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

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Step 2.1.6.2.1

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

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 (p^{3} p^{1}) + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.6.2.2

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

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{3 + 1} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.6.3

Add $3$ and $1$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} + 3 p (- p^{2}) + 3 p \cdot 5$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} + 3 \cdot - 1 p \cdot p^{2} + 3 p \cdot 5$

Step 2.1.8

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

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Step 2.1.8.1

Move $p^{2}$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} + 3 \cdot - 1 (p^{2} p) + 3 p \cdot 5$

Step 2.1.8.2

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

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Step 2.1.8.2.1

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

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} + 3 \cdot - 1 (p^{2} p^{1}) + 3 p \cdot 5$

Step 2.1.8.2.2

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

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} + 3 \cdot - 1 p^{2 + 1} + 3 p \cdot 5$

Step 2.1.8.3

Add $2$ and $1$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} + 3 \cdot - 1 p^{3} + 3 p \cdot 5$

Step 2.1.9

Multiply $3$ by $- 1$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} - 3 p^{3} + 3 p \cdot 5$

Step 2.1.10

Multiply $5$ by $3$ .

$5 p^{5} - 5 p^{4} + 25 p^{2} + 3 p^{4} - 3 p^{3} + 15 p$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 5 p^{4}$ and $3 p^{4}$ .

$5 p^{5} - 2 p^{4} + 25 p^{2} - 3 p^{3} + 15 p$

Step 2.2.2

Move $25 p^{2}$ .

$5 p^{5} - 2 p^{4} - 3 p^{3} + 25 p^{2} + 15 p$