Simplify (1/4m^4-2/3m^3n+3/8m^2n^2)÷(1/4m^2)

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $4$ by $3$ .

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

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

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

Multiply $3$ by $4$ .

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

Combine the numerators over the common denominator.

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

To write $\frac{m^{2} \cdot 3 - 2 n m \cdot 4}{12}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

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

Multiply $\frac{m^{2} \cdot 3 - 2 n m \cdot 4}{12}$ by $\frac{2}{2}$ .

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

Multiply $12$ by $2$ .

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

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

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

Multiply $8$ by $3$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{(m^{2} \cdot 3 - 2 n m \cdot 4) \cdot 2 + 3 n^{2} \cdot 3}{24}$ in a factored form.

Simplify each term.

Move $3$ to the left of $m^{2}$ .

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

Multiply $4$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $- 8$ .

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

Multiply $3$ by $3$ .

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

Step 3

Combine fractions.

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

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

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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

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

Cancel the common factors.

Factor $4$ out of $24$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1/4m^4-2/3m^3n+3/8m^2n^2)÷(1/4m^2)

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