Algebra Examples

$(- \frac{1}{4} y^{2 n} z^{5 n} - \frac{7}{8} y^{4 n} z^{2 n} - \frac{7}{4} y^{4 n}) \div (- \frac{3}{8} y^{4 n} z^{n})$

Step 1

Rewrite the division as a fraction.

$\frac{- \frac{1}{4} \cdot (y^{2 n} z^{5 n}) - \frac{7}{8} \cdot (y^{4 n} z^{2 n}) - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot (y^{4 n} z^{n})}$

Step 2

Simplify the numerator.

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Step 2.1

Combine $y^{2 n}$ and $\frac{1}{4}$ .

$\frac{- \frac{y^{2 n}}{4} z^{5 n} - \frac{7}{8} \cdot y^{4 n} z^{2 n} - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.2

Combine $z^{5 n}$ and $\frac{y^{2 n}}{4}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{7}{8} \cdot y^{4 n} z^{2 n} - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.3

Combine $y^{4 n}$ and $\frac{7}{8}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{y^{4 n} \cdot 7}{8} z^{2 n} - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.4

Move $7$ to the left of $y^{4 n}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{7 \cdot y^{4 n}}{8} z^{2 n} - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.5

Combine $z^{2 n}$ and $\frac{7 y^{4 n}}{8}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{z^{2 n} (7 y^{4 n})}{8} - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.6

Move $7$ to the left of $z^{2 n}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{7 \cdot z^{2 n} y^{4 n}}{8} - \frac{7}{4} y^{4 n}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.7

Combine $y^{4 n}$ and $\frac{7}{4}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{7 z^{2 n} y^{4 n}}{8} - \frac{y^{4 n} \cdot 7}{4}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.8

Move $7$ to the left of $y^{4 n}$ .

$\frac{- \frac{z^{5 n} y^{2 n}}{4} - \frac{7 z^{2 n} y^{4 n}}{8} - \frac{7 \cdot y^{4 n}}{4}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9

Factor $- 1$ out of $- \frac{z^{5 n} y^{2 n}}{4} - \frac{7 z^{2 n} y^{4 n}}{8} - \frac{7 y^{4 n}}{4}$ .

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Step 2.9.1

Reorder the expression.

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Step 2.9.1.1

Reorder $z^{5 n}$ and $y^{2 n}$ .

$\frac{- \frac{y^{2 n} z^{5 n}}{4} - \frac{7 z^{2 n} y^{4 n}}{8} - \frac{7 y^{4 n}}{4}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9.1.2

Move $z^{2 n}$ .

$\frac{- \frac{y^{2 n} z^{5 n}}{4} - \frac{7 y^{4 n} z^{2 n}}{8} - \frac{7 y^{4 n}}{4}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9.2

Factor $- 1$ out of $- \frac{y^{2 n} z^{5 n}}{4}$ .

$\frac{- (\frac{y^{2 n} z^{5 n}}{4}) - \frac{7 y^{4 n} z^{2 n}}{8} - \frac{7 y^{4 n}}{4}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9.3

Factor $- 1$ out of $- \frac{7 y^{4 n} z^{2 n}}{8}$ .

$\frac{- (\frac{y^{2 n} z^{5 n}}{4}) - (\frac{7 y^{4 n} z^{2 n}}{8}) - \frac{7 y^{4 n}}{4}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9.4

Factor $- 1$ out of $- \frac{7 y^{4 n}}{4}$ .

$\frac{- (\frac{y^{2 n} z^{5 n}}{4}) - (\frac{7 y^{4 n} z^{2 n}}{8}) - (\frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9.5

Factor $- 1$ out of $- (\frac{y^{2 n} z^{5 n}}{4}) - (\frac{7 y^{4 n} z^{2 n}}{8})$ .

$\frac{- (\frac{y^{2 n} z^{5 n}}{4} + \frac{7 y^{4 n} z^{2 n}}{8}) - (\frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.9.6

Factor $- 1$ out of $- (\frac{y^{2 n} z^{5 n}}{4} + \frac{7 y^{4 n} z^{2 n}}{8}) - (\frac{7 y^{4 n}}{4})$ .

$\frac{- (\frac{y^{2 n} z^{5 n}}{4} + \frac{7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.10

To write $\frac{y^{2 n} z^{5 n}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- (\frac{y^{2 n} z^{5 n}}{4} \cdot \frac{2}{2} + \frac{7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.11

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.1

Multiply $\frac{y^{2 n} z^{5 n}}{4}$ by $\frac{2}{2}$ .

$\frac{- (\frac{y^{2 n} z^{5 n} \cdot 2}{4 \cdot 2} + \frac{7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.11.2

Multiply $4$ by $2$ .

$\frac{- (\frac{y^{2 n} z^{5 n} \cdot 2}{8} + \frac{7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.12

Combine the numerators over the common denominator.

$\frac{- (\frac{y^{2 n} z^{5 n} \cdot 2 + 7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.13

Move $2$ to the left of $y^{2 n} z^{5 n}$ .

$\frac{- (\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.14

To write $\frac{7 y^{4 n}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- (\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n}}{4} \cdot \frac{2}{2})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.15

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.15.1

Multiply $\frac{7 y^{4 n}}{4}$ by $\frac{2}{2}$ .

$\frac{- (\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n} \cdot 2}{4 \cdot 2})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.15.2

Multiply $4$ by $2$ .

$\frac{- (\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n}}{8} + \frac{7 y^{4 n} \cdot 2}{8})}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.16

Combine the numerators over the common denominator.

$\frac{- \frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 7 y^{4 n} \cdot 2}{8}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 2.17

Multiply $2$ by $7$ .

$\frac{- \frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8}}{- \frac{3}{8} \cdot y^{4 n} z^{n}}$

Step 3

Simplify the denominator.

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Step 3.1

Combine $y^{4 n}$ and $\frac{3}{8}$ .

$\frac{- \frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8}}{- \frac{y^{4 n} \cdot 3}{8} z^{n}}$

Step 3.2

Combine $z^{n}$ and $\frac{y^{4 n} \cdot 3}{8}$ .

$\frac{- \frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8}}{- \frac{z^{n} (y^{4 n} \cdot 3)}{8}}$

Step 4

Remove unnecessary parentheses.

$\frac{- \frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8}}{- \frac{z^{n} y^{4 n} \cdot 3}{8}}$

Step 5

Move $3$ to the left of $z^{n} y^{4 n}$ .

$\frac{- \frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8}}{- \frac{3 z^{n} y^{4 n}}{8}}$

Step 6

Dividing two negative values results in a positive value.

$\frac{\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8}}{\frac{3 z^{n} y^{4 n}}{8}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8} \cdot \frac{8}{3 z^{n} y^{4 n}}$

Step 8

Cancel the common factor of $8$ .

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Step 8.1

Cancel the common factor.

$\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{8} \cdot \frac{8}{3 z^{n} y^{4 n}}$

Step 8.2

Rewrite the expression.

$(2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}) \frac{1}{3 z^{n} y^{4 n}}$

Step 9

Multiply $2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}$ by $\frac{1}{3 z^{n} y^{4 n}}$ .

$\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 10

Split the fraction $\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n} + 14 y^{4 n}}{3 z^{n} y^{4 n}}$ into two fractions.

$\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 11

Split the fraction $\frac{2 y^{2 n} z^{5 n} + 7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}}$ into two fractions.

$\frac{2 y^{2 n} z^{5 n}}{3 z^{n} y^{4 n}} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 12

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

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Step 12.1

Factor $y^{4 n}$ out of $2 y^{2 n} z^{5 n}$ .

$\frac{y^{4 n} (2 y^{- 2 n} z^{5 n})}{3 z^{n} y^{4 n}} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

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

$\frac{y^{4 n} (2 y^{- 2 n} z^{5 n})}{y^{4 n} (3 z^{n})} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 12.2.2

Cancel the common factor.

$\frac{y^{4 n} (2 y^{- 2 n} z^{5 n})}{y^{4 n} (3 z^{n})} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 12.2.3

Rewrite the expression.

$\frac{2 y^{- 2 n} z^{5 n}}{3 z^{n}} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 13

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

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Step 13.1

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

$\frac{z^{n} (2 y^{- 2 n} z^{4 n})}{3 z^{n}} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $z^{n}$ out of $3 z^{n}$ .

$\frac{z^{n} (2 y^{- 2 n} z^{4 n})}{z^{n} \cdot 3} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 13.2.2

Cancel the common factor.

$\frac{z^{n} (2 y^{- 2 n} z^{4 n})}{z^{n} \cdot 3} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 13.2.3

Rewrite the expression.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 14

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

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Step 14.1

Cancel the common factor.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{7 y^{4 n} z^{2 n}}{3 z^{n} y^{4 n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 14.2

Rewrite the expression.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{7 z^{2 n}}{3 z^{n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 15

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

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Step 15.1

Factor $z^{n}$ out of $7 z^{2 n}$ .

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{z^{n} (7 z^{n})}{3 z^{n}} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $z^{n}$ out of $3 z^{n}$ .

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{z^{n} (7 z^{n})}{z^{n} \cdot 3} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 15.2.2

Cancel the common factor.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{z^{n} (7 z^{n})}{z^{n} \cdot 3} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 15.2.3

Rewrite the expression.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{7 z^{n}}{3} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 16

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

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Step 16.1

Cancel the common factor.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{7 z^{n}}{3} + \frac{14 y^{4 n}}{3 z^{n} y^{4 n}}$

Step 16.2

Rewrite the expression.

$\frac{2 y^{- 2 n} z^{4 n}}{3} + \frac{7 z^{n}}{3} + \frac{14}{3 z^{n}}$