Algebra Examples

$\frac{12 m^{3} y^{4} + 12 m^{2} y^{3} + 3 m y^{2}}{6 m^{2} y^{2}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $3 m y^{2}$ out of $12 m^{3} y^{4} + 12 m^{2} y^{3} + 3 m y^{2}$ .

Tap for more steps...

Step 1.1.1

Factor $3 m y^{2}$ out of $12 m^{3} y^{4}$ .

$\frac{3 m y^{2} (4 m^{2} y^{2}) + 12 m^{2} y^{3} + 3 m y^{2}}{6 m^{2} y^{2}}$

Step 1.1.2

Factor $3 m y^{2}$ out of $12 m^{2} y^{3}$ .

$\frac{3 m y^{2} (4 m^{2} y^{2}) + 3 m y^{2} (4 m y) + 3 m y^{2}}{6 m^{2} y^{2}}$

Step 1.1.3

Factor $3 m y^{2}$ out of $3 m y^{2}$ .

$\frac{3 m y^{2} (4 m^{2} y^{2}) + 3 m y^{2} (4 m y) + 3 m y^{2} (1)}{6 m^{2} y^{2}}$

Step 1.1.4

Factor $3 m y^{2}$ out of $3 m y^{2} (4 m^{2} y^{2}) + 3 m y^{2} (4 m y)$ .

$\frac{3 m y^{2} (4 m^{2} y^{2} + 4 m y) + 3 m y^{2} (1)}{6 m^{2} y^{2}}$

Step 1.1.5

Factor $3 m y^{2}$ out of $3 m y^{2} (4 m^{2} y^{2} + 4 m y) + 3 m y^{2} (1)$ .

$\frac{3 m y^{2} (4 m^{2} y^{2} + 4 m y + 1)}{6 m^{2} y^{2}}$

Step 1.2

Rewrite $m^{2} y^{2}$ as ${(m y)}^{2}$ .

$\frac{3 m y^{2} (4 {(m y)}^{2} + 4 (m y) + 1)}{6 m^{2} y^{2}}$

Step 1.3

Let $u = m y$ . Substitute $u$ for all occurrences of $m y$ .

$\frac{3 m y^{2} (4 u^{2} + 4 u + 1)}{6 m^{2} y^{2}}$

Step 1.4

Factor using the perfect square rule.

Tap for more steps...

Step 1.4.1

Rewrite $4 u^{2}$ as ${(2 u)}^{2}$ .

$\frac{3 m y^{2} ({(2 u)}^{2} + 4 u + 1)}{6 m^{2} y^{2}}$

Step 1.4.2

Rewrite $1$ as $1^{2}$ .

$\frac{3 m y^{2} ({(2 u)}^{2} + 4 u + 1^{2})}{6 m^{2} y^{2}}$

Step 1.4.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot (2 u) \cdot 1$

Step 1.4.4

Rewrite the polynomial.

$\frac{3 m y^{2} ({(2 u)}^{2} + 2 \cdot (2 u) \cdot 1 + 1^{2})}{6 m^{2} y^{2}}$

Step 1.4.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 u$ and $b = 1$ .

$\frac{3 m y^{2} {(2 u + 1)}^{2}}{6 m^{2} y^{2}}$

Step 1.5

Replace all occurrences of $u$ with $m y$ .

$\frac{3 m y^{2} {(2 m y + 1)}^{2}}{6 m^{2} y^{2}}$

Step 2

Cancel the common factor of $3$ and $6$ .

Tap for more steps...

Step 2.1

Factor $3$ out of $3 m y^{2} {(2 m y + 1)}^{2}$ .

$\frac{3 (m y^{2} {(2 m y + 1)}^{2})}{6 m^{2} y^{2}}$

Step 2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1

Factor $3$ out of $6 m^{2} y^{2}$ .

$\frac{3 (m y^{2} {(2 m y + 1)}^{2})}{3 (2 m^{2} y^{2})}$

Step 2.2.2

Cancel the common factor.

$\frac{3 (m y^{2} {(2 m y + 1)}^{2})}{3 (2 m^{2} y^{2})}$

Step 2.2.3

Rewrite the expression.

$\frac{m y^{2} {(2 m y + 1)}^{2}}{2 m^{2} y^{2}}$

Step 3

Cancel the common factor of $m$ and $m^{2}$ .

Tap for more steps...

Step 3.1

Factor $m$ out of $m y^{2} {(2 m y + 1)}^{2}$ .

$\frac{m (y^{2} {(2 m y + 1)}^{2})}{2 m^{2} y^{2}}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

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

$\frac{m (y^{2} {(2 m y + 1)}^{2})}{m (2 m y^{2})}$

Step 3.2.2

Cancel the common factor.

$\frac{m (y^{2} {(2 m y + 1)}^{2})}{m (2 m y^{2})}$

Step 3.2.3

Rewrite the expression.

$\frac{y^{2} {(2 m y + 1)}^{2}}{2 m y^{2}}$

Step 4

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{y^{2} {(2 m y + 1)}^{2}}{2 m y^{2}}$

Step 4.2

Rewrite the expression.

$\frac{{(2 m y + 1)}^{2}}{2 m}$

Step 5

Rewrite ${(2 m y + 1)}^{2}$ as $(2 m y + 1) (2 m y + 1)$ .

$\frac{(2 m y + 1) (2 m y + 1)}{2 m}$

Step 6

Expand $(2 m y + 1) (2 m y + 1)$ using the FOIL Method.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{2 m y (2 m y + 1) + 1 (2 m y + 1)}{2 m}$

Step 6.2

Apply the distributive property.

$\frac{2 m y (2 m y) + 2 m y \cdot 1 + 1 (2 m y + 1)}{2 m}$

Step 6.3

Apply the distributive property.

$\frac{2 m y (2 m y) + 2 m y \cdot 1 + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7

Simplify and combine like terms.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply $m$ by $m$ by adding the exponents.

Tap for more steps...

Step 7.1.1.1

Move $m$ .

$\frac{2 (m \cdot m) y (2 y) + 2 m y \cdot 1 + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7.1.1.2

Multiply $m$ by $m$ .

$\frac{2 m^{2} y (2 y) + 2 m y \cdot 1 + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7.1.2

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

Tap for more steps...

Step 7.1.2.1

Move $y$ .

$\frac{2 m^{2} (y \cdot y) \cdot 2 + 2 m y \cdot 1 + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7.1.2.2

Multiply $y$ by $y$ .

$\frac{2 m^{2} y^{2} \cdot 2 + 2 m y \cdot 1 + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7.1.3

Multiply $2$ by $2$ .

$\frac{4 m^{2} y^{2} + 2 m y \cdot 1 + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7.1.4

Multiply $2$ by $1$ .

$\frac{4 m^{2} y^{2} + 2 m y + 1 (2 m y) + 1 \cdot 1}{2 m}$

Step 7.1.5

Multiply $2 m y$ by $1$ .

$\frac{4 m^{2} y^{2} + 2 m y + 2 m y + 1 \cdot 1}{2 m}$

Step 7.1.6

Multiply $1$ by $1$ .

$\frac{4 m^{2} y^{2} + 2 m y + 2 m y + 1}{2 m}$

Step 7.2

Add $2 m y$ and $2 m y$ .

$\frac{4 m^{2} y^{2} + 4 m y + 1}{2 m}$

Step 8

Split the fraction $\frac{4 m^{2} y^{2} + 4 m y + 1}{2 m}$ into two fractions.

$\frac{4 m^{2} y^{2} + 4 m y}{2 m} + \frac{1}{2 m}$

Step 9

Split the fraction $\frac{4 m^{2} y^{2} + 4 m y}{2 m}$ into two fractions.

$\frac{4 m^{2} y^{2}}{2 m} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 10

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 10.1

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

$\frac{2 (2 m^{2} y^{2})}{2 m} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 10.2

Cancel the common factors.

Tap for more steps...

Step 10.2.1

Factor $2$ out of $2 m$ .

$\frac{2 (2 m^{2} y^{2})}{2 (m)} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 10.2.2

Cancel the common factor.

$\frac{2 (2 m^{2} y^{2})}{2 m} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 10.2.3

Rewrite the expression.

$\frac{2 m^{2} y^{2}}{m} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 11

Cancel the common factor of $m^{2}$ and $m$ .

Tap for more steps...

Step 11.1

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

$\frac{m (2 m y^{2})}{m} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 11.2

Cancel the common factors.

Tap for more steps...

Step 11.2.1

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

$\frac{m (2 m y^{2})}{m^{1}} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 11.2.2

Factor $m$ out of $m^{1}$ .

$\frac{m (2 m y^{2})}{m \cdot 1} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 11.2.3

Cancel the common factor.

$\frac{m (2 m y^{2})}{m \cdot 1} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 11.2.4

Rewrite the expression.

$\frac{2 m y^{2}}{1} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 11.2.5

Divide $2 m y^{2}$ by $1$ .

$2 m y^{2} + \frac{4 m y}{2 m} + \frac{1}{2 m}$

Step 12

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 12.1

Factor $2$ out of $4 m y$ .

$2 m y^{2} + \frac{2 (2 m y)}{2 m} + \frac{1}{2 m}$

Step 12.2

Cancel the common factors.

Tap for more steps...

Step 12.2.1

Factor $2$ out of $2 m$ .

$2 m y^{2} + \frac{2 (2 m y)}{2 (m)} + \frac{1}{2 m}$

Step 12.2.2

Cancel the common factor.

$2 m y^{2} + \frac{2 (2 m y)}{2 m} + \frac{1}{2 m}$

Step 12.2.3

Rewrite the expression.

$2 m y^{2} + \frac{2 m y}{m} + \frac{1}{2 m}$

Step 13

Cancel the common factor of $m$ .

Tap for more steps...

Step 13.1

Cancel the common factor.

$2 m y^{2} + \frac{2 m y}{m} + \frac{1}{2 m}$

Step 13.2

Divide $2 y$ by $1$ .

$2 m y^{2} + 2 y + \frac{1}{2 m}$