Basic Math Examples

$\frac{c^{2} y}{5 c^{2} - 26 c y + 5 y^{2}} \div \frac{{(5 c y^{2})}^{2}}{5 c^{2} y - c y^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{c^{2} y}{5 c^{2} - 26 c y + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor by grouping.

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Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 5 = 25$ and whose sum is $b = - 26$ .

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Step 2.1.1.1

Reorder terms.

$\frac{c^{2} y}{5 c^{2} + 5 y^{2} - 26 c y} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.1.2

Reorder $5 y^{2}$ and $- 26 c y$ .

$\frac{c^{2} y}{5 c^{2} - 26 c y + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.1.3

Factor $- 26$ out of $- 26 c y$ .

$\frac{c^{2} y}{5 c^{2} - 26 (c y) + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.1.4

Rewrite $- 26$ as $- 1$ plus $- 25$

$\frac{c^{2} y}{5 c^{2} + (- 1 - 25) (c y) + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.1.5

Apply the distributive property.

$\frac{c^{2} y}{5 c^{2} - 1 (c y) - 25 (c y) + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.1.6

Move parentheses.

$\frac{c^{2} y}{5 c^{2} - 1 c y - 25 c y + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$\frac{c^{2} y}{(5 c^{2} - 1 c y) - 25 c y + 5 y^{2}} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{c^{2} y}{c (5 c - 1 y) - 5 y (5 c - y)} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 c - 1 y$ .

$\frac{c^{2} y}{(5 c - 1 y) (c - 5 y)} \cdot \frac{5 c^{2} y - c y^{2}}{{(5 c y^{2})}^{2}}$

Step 2.2

Rewrite $- 1 y$ as $- y$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Factor $c y$ out of $5 c^{2} y - c y^{2}$ .

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Step 3.1.1

Factor $c y$ out of $5 c^{2} y$ .

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

Step 3.1.2

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

$\frac{c^{2} y}{(5 c - y) (c - 5 y)} \cdot \frac{c y (5 c) + c y (- 1 y)}{{(5 c y^{2})}^{2}}$

Step 3.1.3

Factor $c y$ out of $c y (5 c) + c y (- 1 y)$ .

$\frac{c^{2} y}{(5 c - y) (c - 5 y)} \cdot \frac{c y (5 c - 1 y)}{{(5 c y^{2})}^{2}}$

Step 3.2

Rewrite $- 1 y$ as $- y$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $5 c y^{2}$ .

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

Step 4.2

Apply the product rule to $5 c$ .

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

Step 4.3

Raise $5$ to the power of $2$ .

$\frac{c^{2} y}{(5 c - y) (c - 5 y)} \cdot \frac{c y (5 c - y)}{25 c^{2} {(y^{2})}^{2}}$

Step 4.4

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 4.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{c^{2} y}{(5 c - y) (c - 5 y)} \cdot \frac{c y (5 c - y)}{25 c^{2} y^{2 \cdot 2}}$

Step 4.4.2

Multiply $2$ by $2$ .

$\frac{c^{2} y}{(5 c - y) (c - 5 y)} \cdot \frac{c y (5 c - y)}{25 c^{2} y^{4}}$

Step 5

Combine.

$\frac{c^{2} y (c y (5 c - y))}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 6

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

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Step 6.1

Move $c$ .

$\frac{c \cdot c^{2} y (y (5 c - y))}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 6.2

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

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Step 6.2.1

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

$\frac{c^{1} c^{2} y (y (5 c - y))}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 6.2.2

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

$\frac{c^{1 + 2} y (y (5 c - y))}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 6.3

Add $1$ and $2$ .

$\frac{c^{3} y (y (5 c - y))}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 7

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

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Step 7.1

Move $y$ .

$\frac{c^{3} (y \cdot y) (5 c - y)}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 7.2

Multiply $y$ by $y$ .

$\frac{c^{3} y^{2} (5 c - y)}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 8

Cancel the common factor of $c^{3}$ and $c^{2}$ .

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Step 8.1

Factor $c^{2}$ out of $c^{3} y^{2} (5 c - y)$ .

$\frac{c^{2} (c y^{2} (5 c - y))}{(5 c - y) (c - 5 y) (25 c^{2} y^{4})}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $c^{2}$ out of $(5 c - y) (c - 5 y) (25 c^{2} y^{4})$ .

$\frac{c^{2} (c y^{2} (5 c - y))}{c^{2} ((5 c - y) (c - 5 y) (25 y^{4}))}$

Step 8.2.2

Cancel the common factor.

$\frac{c^{2} (c y^{2} (5 c - y))}{c^{2} ((5 c - y) (c - 5 y) (25 y^{4}))}$

Step 8.2.3

Rewrite the expression.

$\frac{c y^{2} (5 c - y)}{(5 c - y) (c - 5 y) (25 y^{4})}$

Step 9

Cancel the common factor of $y^{2}$ and $y^{4}$ .

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Step 9.1

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

$\frac{y^{2} (c (5 c - y))}{(5 c - y) (c - 5 y) (25 y^{4})}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $y^{2}$ out of $(5 c - y) (c - 5 y) (25 y^{4})$ .

$\frac{y^{2} (c (5 c - y))}{y^{2} ((5 c - y) (c - 5 y) (25 y^{2}))}$

Step 9.2.2

Cancel the common factor.

$\frac{y^{2} (c (5 c - y))}{y^{2} ((5 c - y) (c - 5 y) (25 y^{2}))}$

Step 9.2.3

Rewrite the expression.

$\frac{c (5 c - y)}{(5 c - y) (c - 5 y) (25 y^{2})}$

Step 10

Cancel the common factor of $5 c - y$ .

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Step 10.1

Cancel the common factor.

$\frac{c (5 c - y)}{(5 c - y) (c - 5 y) (25 y^{2})}$

Step 10.2

Rewrite the expression.

$\frac{c}{(c - 5 y) (25 y^{2})}$

Step 11

Simplify the expression.

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Step 11.1

Move $25$ to the left of $c - 5 y$ .

$\frac{c}{25 \cdot (c - 5 y) y^{2}}$

Step 11.2

Reorder factors in $\frac{c}{25 (c - 5 y) y^{2}}$ .

$\frac{c}{25 y^{2} (c - 5 y)}$