Basic Math Examples

$\frac{20 c^{2} y + 4 c y^{2}}{{(3 c - 2 y)}^{2} + {(2 c + 3 y)}^{2}}$

Step 1

Factor $4 c y$ out of $20 c^{2} y + 4 c y^{2}$ .

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Step 1.1

Factor $4 c y$ out of $20 c^{2} y$ .

$\frac{4 c y (5 c) + 4 c y^{2}}{{(3 c - 2 y)}^{2} + {(2 c + 3 y)}^{2}}$

Step 1.2

Factor $4 c y$ out of $4 c y^{2}$ .

$\frac{4 c y (5 c) + 4 c y (y)}{{(3 c - 2 y)}^{2} + {(2 c + 3 y)}^{2}}$

Step 1.3

Factor $4 c y$ out of $4 c y (5 c) + 4 c y (y)$ .

$\frac{4 c y (5 c + y)}{{(3 c - 2 y)}^{2} + {(2 c + 3 y)}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite ${(3 c - 2 y)}^{2}$ as $(3 c - 2 y) (3 c - 2 y)$ .

$\frac{4 c y (5 c + y)}{(3 c - 2 y) (3 c - 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.2

Expand $(3 c - 2 y) (3 c - 2 y)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\frac{4 c y (5 c + y)}{3 c (3 c - 2 y) - 2 y (3 c - 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.2.2

Apply the distributive property.

$\frac{4 c y (5 c + y)}{3 c (3 c) + 3 c (- 2 y) - 2 y (3 c - 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.2.3

Apply the distributive property.

$\frac{4 c y (5 c + y)}{3 c (3 c) + 3 c (- 2 y) - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{3 \cdot 3 c \cdot c + 3 c (- 2 y) - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.2

Multiply $c$ by $c$ by adding the exponents.

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Step 2.3.1.2.1

Move $c$ .

$\frac{4 c y (5 c + y)}{3 \cdot 3 (c \cdot c) + 3 c (- 2 y) - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.2.2

Multiply $c$ by $c$ .

$\frac{4 c y (5 c + y)}{3 \cdot 3 c^{2} + 3 c (- 2 y) - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.3

Multiply $3$ by $3$ .

$\frac{4 c y (5 c + y)}{9 c^{2} + 3 c (- 2 y) - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} + 3 \cdot - 2 c y - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.5

Multiply $3$ by $- 2$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 2 y (3 c) - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.6

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 2 \cdot 3 y c - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.7

Multiply $- 2$ by $3$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 6 y c - 2 y (- 2 y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.8

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 6 y c - 2 \cdot - 2 y \cdot y + {(2 c + 3 y)}^{2}}$

Step 2.3.1.9

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

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Step 2.3.1.9.1

Move $y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 6 y c - 2 \cdot - 2 (y \cdot y) + {(2 c + 3 y)}^{2}}$

Step 2.3.1.9.2

Multiply $y$ by $y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 6 y c - 2 \cdot - 2 y^{2} + {(2 c + 3 y)}^{2}}$

Step 2.3.1.10

Multiply $- 2$ by $- 2$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 6 y c + 4 y^{2} + {(2 c + 3 y)}^{2}}$

Step 2.3.2

Subtract $6 y c$ from $- 6 c y$ .

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Step 2.3.2.1

Move $y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 6 c y - 6 c y + 4 y^{2} + {(2 c + 3 y)}^{2}}$

Step 2.3.2.2

Subtract $6 c y$ from $- 6 c y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + {(2 c + 3 y)}^{2}}$

Step 2.4

Rewrite ${(2 c + 3 y)}^{2}$ as $(2 c + 3 y) (2 c + 3 y)$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + (2 c + 3 y) (2 c + 3 y)}$

Step 2.5

Expand $(2 c + 3 y) (2 c + 3 y)$ using the FOIL Method.

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Step 2.5.1

Apply the distributive property.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 2 c (2 c + 3 y) + 3 y (2 c + 3 y)}$

Step 2.5.2

Apply the distributive property.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 2 c (2 c) + 2 c (3 y) + 3 y (2 c + 3 y)}$

Step 2.5.3

Apply the distributive property.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 2 c (2 c) + 2 c (3 y) + 3 y (2 c) + 3 y (3 y)}$

Step 2.6

Simplify and combine like terms.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 2 \cdot 2 c \cdot c + 2 c (3 y) + 3 y (2 c) + 3 y (3 y)}$

Step 2.6.1.2

Multiply $c$ by $c$ by adding the exponents.

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Step 2.6.1.2.1

Move $c$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 2 \cdot 2 (c \cdot c) + 2 c (3 y) + 3 y (2 c) + 3 y (3 y)}$

Step 2.6.1.2.2

Multiply $c$ by $c$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 2 \cdot 2 c^{2} + 2 c (3 y) + 3 y (2 c) + 3 y (3 y)}$

Step 2.6.1.3

Multiply $2$ by $2$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 2 c (3 y) + 3 y (2 c) + 3 y (3 y)}$

Step 2.6.1.4

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 2 \cdot 3 c y + 3 y (2 c) + 3 y (3 y)}$

Step 2.6.1.5

Multiply $2$ by $3$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 3 y (2 c) + 3 y (3 y)}$

Step 2.6.1.6

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 3 \cdot 2 y c + 3 y (3 y)}$

Step 2.6.1.7

Multiply $3$ by $2$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 6 y c + 3 y (3 y)}$

Step 2.6.1.8

Rewrite using the commutative property of multiplication.

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 6 y c + 3 \cdot 3 y \cdot y}$

Step 2.6.1.9

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

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Step 2.6.1.9.1

Move $y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 6 y c + 3 \cdot 3 (y \cdot y)}$

Step 2.6.1.9.2

Multiply $y$ by $y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 6 y c + 3 \cdot 3 y^{2}}$

Step 2.6.1.10

Multiply $3$ by $3$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 6 y c + 9 y^{2}}$

Step 2.6.2

Add $6 c y$ and $6 y c$ .

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Step 2.6.2.1

Move $y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 6 c y + 6 c y + 9 y^{2}}$

Step 2.6.2.2

Add $6 c y$ and $6 c y$ .

$\frac{4 c y (5 c + y)}{9 c^{2} - 12 c y + 4 y^{2} + 4 c^{2} + 12 c y + 9 y^{2}}$

Step 2.7

Add $9 c^{2}$ and $4 c^{2}$ .

$\frac{4 c y (5 c + y)}{13 c^{2} - 12 c y + 4 y^{2} + 12 c y + 9 y^{2}}$

Step 2.8

Add $- 12 c y$ and $12 c y$ .

$\frac{4 c y (5 c + y)}{13 c^{2} + 4 y^{2} + 0 + 9 y^{2}}$

Step 2.9

Add $13 c^{2}$ and $0$ .

$\frac{4 c y (5 c + y)}{13 c^{2} + 4 y^{2} + 9 y^{2}}$

Step 2.10

Add $4 y^{2}$ and $9 y^{2}$ .

$\frac{4 c y (5 c + y)}{13 c^{2} + 13 y^{2}}$

Step 2.11

Factor $13$ out of $13 c^{2} + 13 y^{2}$ .

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Step 2.11.1

Factor $13$ out of $13 c^{2}$ .

$\frac{4 c y (5 c + y)}{13 (c^{2}) + 13 y^{2}}$

Step 2.11.2

Factor $13$ out of $13 y^{2}$ .

$\frac{4 c y (5 c + y)}{13 (c^{2}) + 13 (y^{2})}$

Step 2.11.3

Factor $13$ out of $13 (c^{2}) + 13 (y^{2})$ .

$\frac{4 c y (5 c + y)}{13 (c^{2} + y^{2})}$