Basic Math 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

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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 2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.1

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 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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 2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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