Pre-Algebra Examples

$(9 x^{6} y - 25 x^{6} y^{7}) \div (- 3 x^{2} y^{5})$

Step 1

Rewrite the division as a fraction.

$\frac{9 x^{6} y - 25 x^{6} y^{7}}{- 3 x^{2} y^{5}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $x^{6} y$ out of $9 x^{6} y - 25 x^{6} y^{7}$ .

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Step 2.1.1

Factor $x^{6} y$ out of $9 x^{6} y$ .

$\frac{x^{6} y (9) - 25 x^{6} y^{7}}{- 3 x^{2} y^{5}}$

Step 2.1.2

Factor $x^{6} y$ out of $- 25 x^{6} y^{7}$ .

$\frac{x^{6} y (9) + x^{6} y (- 25 y^{6})}{- 3 x^{2} y^{5}}$

Step 2.1.3

Factor $x^{6} y$ out of $x^{6} y (9) + x^{6} y (- 25 y^{6})$ .

$\frac{x^{6} y (9 - 25 y^{6})}{- 3 x^{2} y^{5}}$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{x^{6} y (3^{2} - 25 y^{6})}{- 3 x^{2} y^{5}}$

Step 2.3

Rewrite $25 y^{6}$ as ${(5 y^{3})}^{2}$ .

$\frac{x^{6} y (3^{2} - {(5 y^{3})}^{2})}{- 3 x^{2} y^{5}}$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = 5 y^{3}$ .

$\frac{x^{6} y ((3 + 5 y^{3}) (3 - (5 y^{3})))}{- 3 x^{2} y^{5}}$

Step 2.5

Multiply $5$ by $- 1$ .

$\frac{x^{6} y (3 + 5 y^{3}) (3 - 5 y^{3})}{- 3 x^{2} y^{5}}$

Step 3

Cancel the common factor of $x^{6}$ and $x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $x^{6} y (3 + 5 y^{3}) (3 - 5 y^{3})$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{2}$ out of $- 3 x^{2} y^{5}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Cancel the common factor of $y$ and $y^{5}$ .

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Step 4.1

Factor $y$ out of $(x^{4} y (3 + 5 y^{3})) (3 - 5 y^{3})$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $y$ out of $- 3 y^{5}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Move the negative in front of the fraction.

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

Step 6

Apply the distributive property.

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

Step 7

Move $3$ to the left of $x^{4}$ .

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

Step 8

Rewrite using the commutative property of multiplication.

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

Step 9

Expand $(3 x^{4} + 5 x^{4} y^{3}) (3 - 5 y^{3})$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply $3$ by $3$ .

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

Step 10.1.2

Rewrite using the commutative property of multiplication.

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

Step 10.1.3

Multiply $3$ by $- 5$ .

$- \frac{9 x^{4} - 15 x^{4} y^{3} + 5 x^{4} y^{3} \cdot 3 + 5 x^{4} y^{3} (- 5 y^{3})}{3 y^{4}}$

Step 10.1.4

Multiply $3$ by $5$ .

$- \frac{9 x^{4} - 15 x^{4} y^{3} + 15 x^{4} y^{3} + 5 x^{4} y^{3} (- 5 y^{3})}{3 y^{4}}$

Step 10.1.5

Multiply $y^{3}$ by $y^{3}$ by adding the exponents.

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Step 10.1.5.1

Move $y^{3}$ .

$- \frac{9 x^{4} - 15 x^{4} y^{3} + 15 x^{4} y^{3} + 5 x^{4} (y^{3} y^{3}) \cdot - 5}{3 y^{4}}$

Step 10.1.5.2

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

$- \frac{9 x^{4} - 15 x^{4} y^{3} + 15 x^{4} y^{3} + 5 x^{4} y^{3 + 3} \cdot - 5}{3 y^{4}}$

Step 10.1.5.3

Add $3$ and $3$ .

$- \frac{9 x^{4} - 15 x^{4} y^{3} + 15 x^{4} y^{3} + 5 x^{4} y^{6} \cdot - 5}{3 y^{4}}$

Step 10.1.6

Multiply $- 5$ by $5$ .

$- \frac{9 x^{4} - 15 x^{4} y^{3} + 15 x^{4} y^{3} - 25 x^{4} y^{6}}{3 y^{4}}$

Step 10.2

Add $- 15 x^{4} y^{3}$ and $15 x^{4} y^{3}$ .

$- \frac{9 x^{4} + 0 - 25 x^{4} y^{6}}{3 y^{4}}$

Step 10.3

Add $9 x^{4}$ and $0$ .

$- \frac{9 x^{4} - 25 x^{4} y^{6}}{3 y^{4}}$

Step 11

Split the fraction $\frac{9 x^{4} - 25 x^{4} y^{6}}{3 y^{4}}$ into two fractions.

$- (\frac{9 x^{4}}{3 y^{4}} + \frac{- 25 x^{4} y^{6}}{3 y^{4}})$

Step 12

Cancel the common factor of $9$ and $3$ .

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Step 12.1

Factor $3$ out of $9 x^{4}$ .

$- (\frac{3 (3 x^{4})}{3 y^{4}} + \frac{- 25 x^{4} y^{6}}{3 y^{4}})$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $3$ out of $3 y^{4}$ .

$- (\frac{3 (3 x^{4})}{3 (y^{4})} + \frac{- 25 x^{4} y^{6}}{3 y^{4}})$

Step 12.2.2

Cancel the common factor.

$- (\frac{3 (3 x^{4})}{3 y^{4}} + \frac{- 25 x^{4} y^{6}}{3 y^{4}})$

Step 12.2.3

Rewrite the expression.

$- (\frac{3 x^{4}}{y^{4}} + \frac{- 25 x^{4} y^{6}}{3 y^{4}})$

Step 13

Cancel the common factor of $y^{6}$ and $y^{4}$ .

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Step 13.1

Factor $y^{4}$ out of $- 25 x^{4} y^{6}$ .

$- (\frac{3 x^{4}}{y^{4}} + \frac{y^{4} (- 25 x^{4} y^{2})}{3 y^{4}})$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $y^{4}$ out of $3 y^{4}$ .

$- (\frac{3 x^{4}}{y^{4}} + \frac{y^{4} (- 25 x^{4} y^{2})}{y^{4} \cdot 3})$

Step 13.2.2

Cancel the common factor.

$- (\frac{3 x^{4}}{y^{4}} + \frac{y^{4} (- 25 x^{4} y^{2})}{y^{4} \cdot 3})$

Step 13.2.3

Rewrite the expression.

$- (\frac{3 x^{4}}{y^{4}} + \frac{- 25 x^{4} y^{2}}{3})$

Step 14

Move the negative in front of the fraction.

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