Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Simplify the numerator.

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Step 2.1

Factor $m^{2} n$ out of $\frac{1}{7} \cdot m^{3} n^{2} - \frac{2}{5} \cdot m^{4} n + \frac{3}{8} \cdot m^{2} n^{2}$ .

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Step 2.1.1

Factor $m^{2} n$ out of $\frac{1}{7} \cdot m^{3} n^{2}$ .

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

Step 2.1.2

Factor $m^{2} n$ out of $- \frac{2}{5} \cdot m^{4} n$ .

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

Step 2.1.3

Factor $m^{2} n$ out of $\frac{3}{8} \cdot m^{2} n^{2}$ .

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

Step 2.1.4

Factor $m^{2} n$ out of $m^{2} n (\frac{1}{7} \cdot m n) + m^{2} n (- \frac{2}{5} \cdot m^{2})$ .

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

Step 2.1.5

Factor $m^{2} n$ out of $m^{2} n (\frac{1}{7} \cdot m n - \frac{2}{5} \cdot m^{2}) + m^{2} n (\frac{3}{8} n)$ .

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

Step 2.2

Combine exponents.

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Step 2.2.1

Combine $\frac{1}{7}$ and $m$ .

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

Step 2.2.2

Combine $\frac{m}{7}$ and $n$ .

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

Step 2.2.3

Combine $m^{2}$ and $\frac{2}{5}$ .

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

Step 2.2.4

Combine $\frac{3}{8}$ and $n$ .

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

Step 2.3

Move $2$ to the left of $m^{2}$ .

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

Step 2.4

To write $\frac{m n}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.5

To write $- \frac{2 m^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

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

Step 2.6

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.1

Multiply $\frac{m n}{7}$ by $\frac{5}{5}$ .

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

Step 2.6.2

Multiply $7$ by $5$ .

$\frac{m^{2} n (\frac{m n \cdot 5}{35} - \frac{2 m^{2}}{5} \cdot \frac{7}{7} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.6.3

Multiply $\frac{2 m^{2}}{5}$ by $\frac{7}{7}$ .

$\frac{m^{2} n (\frac{m n \cdot 5}{35} - \frac{2 m^{2} \cdot 7}{5 \cdot 7} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.6.4

Multiply $5$ by $7$ .

$\frac{m^{2} n (\frac{m n \cdot 5}{35} - \frac{2 m^{2} \cdot 7}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{m^{2} n (\frac{m n \cdot 5 - 2 m^{2} \cdot 7}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Factor $m$ out of $m n \cdot 5 - 2 m^{2} \cdot 7$ .

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Step 2.8.1.1

Factor $m$ out of $m n \cdot 5$ .

$\frac{m^{2} n (\frac{m (n \cdot 5) - 2 m^{2} \cdot 7}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.8.1.2

Factor $m$ out of $- 2 m^{2} \cdot 7$ .

$\frac{m^{2} n (\frac{m (n \cdot 5) + m (- 2 m \cdot 7)}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.8.1.3

Factor $m$ out of $m (n \cdot 5) + m (- 2 m \cdot 7)$ .

$\frac{m^{2} n (\frac{m (n \cdot 5 - 2 m \cdot 7)}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.8.2

Move $5$ to the left of $n$ .

$\frac{m^{2} n (\frac{m (5 \cdot n - 2 m \cdot 7)}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.8.3

Multiply $7$ by $- 2$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m)}{35} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.9

To write $\frac{m (5 n - 14 m)}{35}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m)}{35} \cdot \frac{8}{8} + \frac{3 n}{8})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.10

To write $\frac{3 n}{8}$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m)}{35} \cdot \frac{8}{8} + \frac{3 n}{8} \cdot \frac{35}{35})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.11

Write each expression with a common denominator of $280$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.1

Multiply $\frac{m (5 n - 14 m)}{35}$ by $\frac{8}{8}$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m) \cdot 8}{35 \cdot 8} + \frac{3 n}{8} \cdot \frac{35}{35})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.11.2

Multiply $35$ by $8$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m) \cdot 8}{280} + \frac{3 n}{8} \cdot \frac{35}{35})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.11.3

Multiply $\frac{3 n}{8}$ by $\frac{35}{35}$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m) \cdot 8}{280} + \frac{3 n \cdot 35}{8 \cdot 35})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.11.4

Multiply $8$ by $35$ .

$\frac{m^{2} n (\frac{m (5 n - 14 m) \cdot 8}{280} + \frac{3 n \cdot 35}{280})}{\frac{1}{4} \cdot m^{2} n}$

Step 2.12

Combine the numerators over the common denominator.

$\frac{m^{2} n \frac{m (5 n - 14 m) \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Apply the distributive property.

$\frac{m^{2} n \frac{(m (5 n) + m (- 14 m)) \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.2

Rewrite using the commutative property of multiplication.

$\frac{m^{2} n \frac{(5 m n + m (- 14 m)) \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.3

Rewrite using the commutative property of multiplication.

$\frac{m^{2} n \frac{(5 m n - 14 m \cdot m) \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.4

Multiply $m$ by $m$ by adding the exponents.

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Step 2.13.4.1

Move $m$ .

$\frac{m^{2} n \frac{(5 m n - 14 (m \cdot m)) \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.4.2

Multiply $m$ by $m$ .

$\frac{m^{2} n \frac{(5 m n - 14 m^{2}) \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.5

Apply the distributive property.

$\frac{m^{2} n \frac{5 m n \cdot 8 - 14 m^{2} \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.6

Multiply $8$ by $5$ .

$\frac{m^{2} n \frac{40 m n - 14 m^{2} \cdot 8 + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.7

Multiply $8$ by $- 14$ .

$\frac{m^{2} n \frac{40 m n - 112 m^{2} + 3 n \cdot 35}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.13.8

Multiply $35$ by $3$ .

$\frac{m^{2} n \frac{40 m n - 112 m^{2} + 105 n}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.14

Combine exponents.

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Step 2.14.1

Combine $m^{2}$ and $\frac{40 m n - 112 m^{2} + 105 n}{280}$ .

$\frac{n \frac{m^{2} (40 m n - 112 m^{2} + 105 n)}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.14.2

Combine $n$ and $\frac{m^{2} (40 m n - 112 m^{2} + 105 n)}{280}$ .

$\frac{\frac{n (m^{2} (40 m n - 112 m^{2} + 105 n))}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 2.15

Remove unnecessary parentheses.

$\frac{\frac{n m^{2} (40 m n - 112 m^{2} + 105 n)}{280}}{\frac{1}{4} \cdot m^{2} n}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{\frac{n m^{2} (40 m n - 112 m^{2} + 105 n)}{280}}{\frac{m^{2}}{4} n}$

Step 3.2

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

$\frac{\frac{n m^{2} (40 m n - 112 m^{2} + 105 n)}{280}}{\frac{m^{2} n}{4}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{n m^{2} (40 m n - 112 m^{2} + 105 n)}{280} \cdot \frac{4}{m^{2} n}$

Step 5

Combine.

$\frac{n m^{2} (40 m n - 112 m^{2} + 105 n) \cdot 4}{280 (m^{2} n)}$

Step 6

Cancel the common factor of $n$ .

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Step 6.1

Cancel the common factor.

$\frac{n m^{2} (40 m n - 112 m^{2} + 105 n) \cdot 4}{280 (m^{2} n)}$

Step 6.2

Rewrite the expression.

$\frac{(m^{2} (40 m n - 112 m^{2} + 105 n)) \cdot 4}{280 (m^{2})}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{m^{2} (40 m n - 112 m^{2} + 105 n) \cdot 4}{280 m^{2}}$

Step 7.2

Rewrite the expression.

$\frac{(40 m n - 112 m^{2} + 105 n) \cdot 4}{280}$

Step 8

Cancel the common factor of $4$ and $280$ .

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Step 8.1

Factor $4$ out of $(40 m n - 112 m^{2} + 105 n) \cdot 4$ .

$\frac{4 \cdot (40 m n - 112 m^{2} + 105 n)}{280}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $4$ out of $280$ .

$\frac{4 \cdot (40 m n - 112 m^{2} + 105 n)}{4 \cdot 70}$

Step 8.2.2

Cancel the common factor.

$\frac{4 \cdot (40 m n - 112 m^{2} + 105 n)}{4 \cdot 70}$

Step 8.2.3

Rewrite the expression.

$\frac{40 m n - 112 m^{2} + 105 n}{70}$