Basic Math Examples

$(a y + y) (3 a y^{2} - 5 y)$

Step 1

Expand $(a y + y) (3 a y^{2} - 5 y)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$a y (3 a y^{2} - 5 y) + y (3 a y^{2} - 5 y)$

Step 1.2

Apply the distributive property.

$a y (3 a y^{2}) + a y (- 5 y) + y (3 a y^{2} - 5 y)$

Step 1.3

Apply the distributive property.

$a y (3 a y^{2}) + a y (- 5 y) + y (3 a y^{2}) + y (- 5 y)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$3 a y (a y^{2}) + a y (- 5 y) + y (3 a y^{2}) + y (- 5 y)$

Step 2.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.1.2.1

Move $a$ .

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

Step 2.1.2.2

Multiply $a$ by $a$ .

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

Step 2.1.3

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

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Step 2.1.3.1

Move $y^{2}$ .

$3 a^{2} (y^{2} y) + a y (- 5 y) + y (3 a y^{2}) + y (- 5 y)$

Step 2.1.3.2

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

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Step 2.1.3.2.1

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

$3 a^{2} (y^{2} y^{1}) + a y (- 5 y) + y (3 a y^{2}) + y (- 5 y)$

Step 2.1.3.2.2

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

$3 a^{2} y^{2 + 1} + a y (- 5 y) + y (3 a y^{2}) + y (- 5 y)$

Step 2.1.3.3

Add $2$ and $1$ .

$3 a^{2} y^{3} + a y (- 5 y) + y (3 a y^{2}) + y (- 5 y)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.5.1

Move $y$ .

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

Step 2.1.5.2

Multiply $y$ by $y$ .

$3 a^{2} y^{3} - 5 a y^{2} + y (3 a y^{2}) + y (- 5 y)$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$3 a^{2} y^{3} - 5 a y^{2} + 3 y (a y^{2}) + y (- 5 y)$

Step 2.1.7

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

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Step 2.1.7.1

Move $y^{2}$ .

$3 a^{2} y^{3} - 5 a y^{2} + 3 (y^{2} y) a + y (- 5 y)$

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

$3 a^{2} y^{3} - 5 a y^{2} + 3 (y^{2} y^{1}) a + y (- 5 y)$

Step 2.1.7.2.2

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

$3 a^{2} y^{3} - 5 a y^{2} + 3 y^{2 + 1} a + y (- 5 y)$

Step 2.1.7.3

Add $2$ and $1$ .

$3 a^{2} y^{3} - 5 a y^{2} + 3 y^{3} a + y (- 5 y)$

Step 2.1.8

Rewrite using the commutative property of multiplication.

$3 a^{2} y^{3} - 5 a y^{2} + 3 y^{3} a - 5 y \cdot y$

Step 2.1.9

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.9.1

Move $y$ .

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

Step 2.1.9.2

Multiply $y$ by $y$ .

$3 a^{2} y^{3} - 5 a y^{2} + 3 y^{3} a - 5 y^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $y^{3}$ .

$3 a^{2} y^{3} - 5 a y^{2} + 3 a y^{3} - 5 y^{2}$

Step 2.2.2

Move $- 5 a y^{2}$ .

$3 a^{2} y^{3} + 3 a y^{3} - 5 a y^{2} - 5 y^{2}$