Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Factor $m^{2}$ out of $\frac{1}{4} m^{4}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Factor $m^{2}$ out of $m^{2} (\frac{1}{4} m^{2}) + m^{2} (- \frac{2}{3} \cdot m n)$ .

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

Step 2.1.5

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

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

Step 2.2

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

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

Step 2.3

Combine $m$ and $\frac{2}{3}$ .

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

Step 2.4

Move $2$ to the left of $m$ .

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

Step 2.5

Combine $n$ and $\frac{2 m}{3}$ .

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

Step 2.6

Move $2$ to the left of $n$ .

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

Step 2.7

Combine $\frac{3}{8}$ and $n^{2}$ .

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

Step 2.8

To write $\frac{m^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.9

To write $- \frac{2 n m}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.10

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.1

Multiply $\frac{m^{2}}{4}$ by $\frac{3}{3}$ .

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

Step 2.10.2

Multiply $4$ by $3$ .

$\frac{m^{2} (\frac{m^{2} \cdot 3}{12} - \frac{2 n m}{3} \cdot \frac{4}{4} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.10.3

Multiply $\frac{2 n m}{3}$ by $\frac{4}{4}$ .

$\frac{m^{2} (\frac{m^{2} \cdot 3}{12} - \frac{2 n m \cdot 4}{3 \cdot 4} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.10.4

Multiply $3$ by $4$ .

$\frac{m^{2} (\frac{m^{2} \cdot 3}{12} - \frac{2 n m \cdot 4}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.11

Combine the numerators over the common denominator.

$\frac{m^{2} (\frac{m^{2} \cdot 3 - 2 n m \cdot 4}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.12

Simplify the numerator.

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Step 2.12.1

Factor $m$ out of $m^{2} \cdot 3 - 2 n m \cdot 4$ .

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Step 2.12.1.1

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

$\frac{m^{2} (\frac{m (m \cdot 3) - 2 n m \cdot 4}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.12.1.2

Factor $m$ out of $- 2 n m \cdot 4$ .

$\frac{m^{2} (\frac{m (m \cdot 3) + m (- 2 n \cdot 4)}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.12.1.3

Factor $m$ out of $m (m \cdot 3) + m (- 2 n \cdot 4)$ .

$\frac{m^{2} (\frac{m (m \cdot 3 - 2 n \cdot 4)}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.12.2

Move $3$ to the left of $m$ .

$\frac{m^{2} (\frac{m (3 \cdot m - 2 n \cdot 4)}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.12.3

Multiply $4$ by $- 2$ .

$\frac{m^{2} (\frac{m (3 m - 8 n)}{12} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.13

To write $\frac{m (3 m - 8 n)}{12}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{m^{2} (\frac{m (3 m - 8 n)}{12} \cdot \frac{2}{2} + \frac{3 n^{2}}{8})}{\frac{1}{4} m^{2}}$

Step 2.14

To write $\frac{3 n^{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{m^{2} (\frac{m (3 m - 8 n)}{12} \cdot \frac{2}{2} + \frac{3 n^{2}}{8} \cdot \frac{3}{3})}{\frac{1}{4} m^{2}}$

Step 2.15

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.15.1

Multiply $\frac{m (3 m - 8 n)}{12}$ by $\frac{2}{2}$ .

$\frac{m^{2} (\frac{m (3 m - 8 n) \cdot 2}{12 \cdot 2} + \frac{3 n^{2}}{8} \cdot \frac{3}{3})}{\frac{1}{4} m^{2}}$

Step 2.15.2

Multiply $12$ by $2$ .

$\frac{m^{2} (\frac{m (3 m - 8 n) \cdot 2}{24} + \frac{3 n^{2}}{8} \cdot \frac{3}{3})}{\frac{1}{4} m^{2}}$

Step 2.15.3

Multiply $\frac{3 n^{2}}{8}$ by $\frac{3}{3}$ .

$\frac{m^{2} (\frac{m (3 m - 8 n) \cdot 2}{24} + \frac{3 n^{2} \cdot 3}{8 \cdot 3})}{\frac{1}{4} m^{2}}$

Step 2.15.4

Multiply $8$ by $3$ .

$\frac{m^{2} (\frac{m (3 m - 8 n) \cdot 2}{24} + \frac{3 n^{2} \cdot 3}{24})}{\frac{1}{4} m^{2}}$

Step 2.16

Combine the numerators over the common denominator.

$\frac{m^{2} \frac{m (3 m - 8 n) \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17

Simplify the numerator.

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Step 2.17.1

Apply the distributive property.

$\frac{m^{2} \frac{(m (3 m) + m (- 8 n)) \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.2

Rewrite using the commutative property of multiplication.

$\frac{m^{2} \frac{(3 m \cdot m + m (- 8 n)) \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.3

Rewrite using the commutative property of multiplication.

$\frac{m^{2} \frac{(3 m \cdot m - 8 m n) \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.4

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

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Step 2.17.4.1

Move $m$ .

$\frac{m^{2} \frac{(3 (m \cdot m) - 8 m n) \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.4.2

Multiply $m$ by $m$ .

$\frac{m^{2} \frac{(3 m^{2} - 8 m n) \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.5

Apply the distributive property.

$\frac{m^{2} \frac{3 m^{2} \cdot 2 - 8 m n \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.6

Multiply $2$ by $3$ .

$\frac{m^{2} \frac{6 m^{2} - 8 m n \cdot 2 + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.7

Multiply $2$ by $- 8$ .

$\frac{m^{2} \frac{6 m^{2} - 16 m n + 3 n^{2} \cdot 3}{24}}{\frac{1}{4} m^{2}}$

Step 2.17.8

Multiply $3$ by $3$ .

$\frac{m^{2} \frac{6 m^{2} - 16 m n + 9 n^{2}}{24}}{\frac{1}{4} m^{2}}$

Step 3

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

$\frac{m^{2} \frac{6 m^{2} - 16 m n + 9 n^{2}}{24}}{\frac{m^{2}}{4}}$

Step 4

Combine $m^{2}$ and $\frac{6 m^{2} - 16 m n + 9 n^{2}}{24}$ .

$\frac{\frac{m^{2} (6 m^{2} - 16 m n + 9 n^{2})}{24}}{\frac{m^{2}}{4}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{m^{2} (6 m^{2} - 16 m n + 9 n^{2})}{24} \cdot \frac{4}{m^{2}}$

Step 6

Combine.

$\frac{m^{2} (6 m^{2} - 16 m n + 9 n^{2}) \cdot 4}{24 m^{2}}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{m^{2} (6 m^{2} - 16 m n + 9 n^{2}) \cdot 4}{24 m^{2}}$

Step 7.2

Rewrite the expression.

$\frac{(6 m^{2} - 16 m n + 9 n^{2}) \cdot 4}{24}$

Step 8

Cancel the common factor of $4$ and $24$ .

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Step 8.1

Factor $4$ out of $(6 m^{2} - 16 m n + 9 n^{2}) \cdot 4$ .

$\frac{4 \cdot (6 m^{2} - 16 m n + 9 n^{2})}{24}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $4$ out of $24$ .

$\frac{4 \cdot (6 m^{2} - 16 m n + 9 n^{2})}{4 \cdot 6}$

Step 8.2.2

Cancel the common factor.

$\frac{4 \cdot (6 m^{2} - 16 m n + 9 n^{2})}{4 \cdot 6}$

Step 8.2.3

Rewrite the expression.

$\frac{6 m^{2} - 16 m n + 9 n^{2}}{6}$

Step 9

Split the fraction $\frac{6 m^{2} - 16 m n + 9 n^{2}}{6}$ into two fractions.

$\frac{6 m^{2} - 16 m n}{6} + \frac{9 n^{2}}{6}$

Step 10

Split the fraction $\frac{6 m^{2} - 16 m n}{6}$ into two fractions.

$\frac{6 m^{2}}{6} + \frac{- 16 m n}{6} + \frac{9 n^{2}}{6}$

Step 11

Cancel the common factor of $6$ .

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Step 11.1

Cancel the common factor.

$\frac{6 m^{2}}{6} + \frac{- 16 m n}{6} + \frac{9 n^{2}}{6}$

Step 11.2

Divide $m^{2}$ by $1$ .

$m^{2} + \frac{- 16 m n}{6} + \frac{9 n^{2}}{6}$

Step 12

Cancel the common factor of $- 16$ and $6$ .

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Step 12.1

Factor $2$ out of $- 16 m n$ .

$m^{2} + \frac{2 (- 8 m n)}{6} + \frac{9 n^{2}}{6}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $2$ out of $6$ .

$m^{2} + \frac{2 (- 8 m n)}{2 (3)} + \frac{9 n^{2}}{6}$

Step 12.2.2

Cancel the common factor.

$m^{2} + \frac{2 (- 8 m n)}{2 \cdot 3} + \frac{9 n^{2}}{6}$

Step 12.2.3

Rewrite the expression.

$m^{2} + \frac{- 8 m n}{3} + \frac{9 n^{2}}{6}$

Step 13

Move the negative in front of the fraction.

$m^{2} - \frac{8 m n}{3} + \frac{9 n^{2}}{6}$

Step 14

Cancel the common factor of $9$ and $6$ .

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Step 14.1

Factor $3$ out of $9 n^{2}$ .

$m^{2} - \frac{8 m n}{3} + \frac{3 (3 n^{2})}{6}$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $3$ out of $6$ .

$m^{2} - \frac{8 m n}{3} + \frac{3 (3 n^{2})}{3 (2)}$

Step 14.2.2

Cancel the common factor.

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

Step 14.2.3

Rewrite the expression.

$m^{2} - \frac{8 m n}{3} + \frac{3 n^{2}}{2}$