Finite Math Examples

$\frac{\frac{y^{2} - 16 y + 64}{7 y^{2} - 56 y}}{\frac{y^{2} - 14 y + 48}{35 y^{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 16 y + 64}{7 y^{2} - 56 y} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

Rewrite $64$ as $8^{2}$ .

$\frac{y^{2} - 16 y + 8^{2}}{7 y^{2} - 56 y} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 2.2

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

$16 y = 2 \cdot y \cdot 8$

Step 2.3

Rewrite the polynomial.

$\frac{y^{2} - 2 \cdot y \cdot 8 + 8^{2}}{7 y^{2} - 56 y} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 8$ .

$\frac{{(y - 8)}^{2}}{7 y^{2} - 56 y} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Factor $7 y$ out of $7 y^{2} - 56 y$ .

Tap for more steps...

Step 3.1.1

Factor $7 y$ out of $7 y^{2}$ .

$\frac{{(y - 8)}^{2}}{7 y (y) - 56 y} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 3.1.2

Factor $7 y$ out of $- 56 y$ .

$\frac{{(y - 8)}^{2}}{7 y (y) + 7 y (- 8)} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 3.1.3

Factor $7 y$ out of $7 y (y) + 7 y (- 8)$ .

$\frac{{(y - 8)}^{2}}{7 y (y - 8)} \cdot \frac{35 y^{2}}{y^{2} - 14 y + 48}$

Step 3.2

Cancel the common factor of $7 y$ .

Tap for more steps...

Step 3.2.1

Factor $7 y$ out of $35 y^{2}$ .

$\frac{{(y - 8)}^{2}}{7 y (y - 8)} \cdot \frac{7 y (5 y)}{y^{2} - 14 y + 48}$

Step 3.2.2

Cancel the common factor.

$\frac{{(y - 8)}^{2}}{7 y (y - 8)} \cdot \frac{7 y (5 y)}{y^{2} - 14 y + 48}$

Step 3.2.3

Rewrite the expression.

$\frac{{(y - 8)}^{2}}{y - 8} \cdot \frac{5 y}{y^{2} - 14 y + 48}$

Step 3.3

Multiply $\frac{{(y - 8)}^{2}}{y - 8}$ by $\frac{5 y}{y^{2} - 14 y + 48}$ .

$\frac{{(y - 8)}^{2} (5 y)}{(y - 8) (y^{2} - 14 y + 48)}$

Step 3.4

Cancel the common factor of ${(y - 8)}^{2}$ and $y - 8$ .

Tap for more steps...

Step 3.4.1

Factor $y - 8$ out of ${(y - 8)}^{2} (5 y)$ .

$\frac{(y - 8) ((y - 8) (5 y))}{(y - 8) (y^{2} - 14 y + 48)}$

Step 3.4.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.1

Cancel the common factor.

$\frac{(y - 8) ((y - 8) (5 y))}{(y - 8) (y^{2} - 14 y + 48)}$

Step 3.4.2.2

Rewrite the expression.

$\frac{(y - 8) (5 y)}{y^{2} - 14 y + 48}$

Step 4

Factor $y^{2} - 14 y + 48$ using the AC method.

Tap for more steps...

Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $48$ and whose sum is $- 14$ .

$- 8, - 6$

Step 4.2

Write the factored form using these integers.

$\frac{(y - 8) \cdot 5 y}{(y - 8) (y - 6)}$

Step 5

Cancel the common factor of $y - 8$ .

Tap for more steps...

Step 5.1

Cancel the common factor.

$\frac{(y - 8) \cdot 5 y}{(y - 8) (y - 6)}$

Step 5.2

Rewrite the expression.

$\frac{5 y}{y - 6}$