Algebra Examples

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

Step 1

Combine fractions.

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Step 1.1

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

${(\frac{m}{4} - 2 n)}^{2}$

Step 1.2

Rewrite ${(\frac{m}{4} - 2 n)}^{2}$ as $(\frac{m}{4} - 2 n) (\frac{m}{4} - 2 n)$ .

$(\frac{m}{4} - 2 n) (\frac{m}{4} - 2 n)$

Step 2

Expand $(\frac{m}{4} - 2 n) (\frac{m}{4} - 2 n)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{m}{4} (\frac{m}{4} - 2 n) - 2 n (\frac{m}{4} - 2 n)$

Step 2.2

Apply the distributive property.

$\frac{m}{4} \cdot \frac{m}{4} + \frac{m}{4} (- 2 n) - 2 n (\frac{m}{4} - 2 n)$

Step 2.3

Apply the distributive property.

$\frac{m}{4} \cdot \frac{m}{4} + \frac{m}{4} (- 2 n) - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{m}{4} \cdot \frac{m}{4}$ .

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Step 3.1.1.1

Multiply $\frac{m}{4}$ by $\frac{m}{4}$ .

$\frac{m \cdot m}{4 \cdot 4} + \frac{m}{4} (- 2 n) - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.1.2

Raise $m$ to the power of $1$ .

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

Step 3.1.1.3

Raise $m$ to the power of $1$ .

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.1.6

Multiply $4$ by $4$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\frac{m^{2}}{16} - 2 \frac{m}{4} n - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Factor $2$ out of $- 2$ .

$\frac{m^{2}}{16} + 2 (- 1) \frac{m}{4} n - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.3.2

Factor $2$ out of $4$ .

$\frac{m^{2}}{16} + 2 \cdot - 1 \frac{m}{2 \cdot 2} n - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.3.3

Cancel the common factor.

$\frac{m^{2}}{16} + 2 \cdot - 1 \frac{m}{2 \cdot 2} n - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.3.4

Rewrite the expression.

$\frac{m^{2}}{16} - \frac{m}{2} n - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.4

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

$\frac{m^{2}}{16} - \frac{n m}{2} - 2 n \frac{m}{4} - 2 n (- 2 n)$

Step 3.1.5

Cancel the common factor of $2$ .

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Step 3.1.5.1

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

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

Step 3.1.5.2

Factor $2$ out of $4$ .

$\frac{m^{2}}{16} - \frac{n m}{2} + 2 (- n) \frac{m}{2 (2)} - 2 n (- 2 n)$

Step 3.1.5.3

Cancel the common factor.

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

Step 3.1.5.4

Rewrite the expression.

$\frac{m^{2}}{16} - \frac{n m}{2} - n \frac{m}{2} - 2 n (- 2 n)$

Step 3.1.6

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

$\frac{m^{2}}{16} - \frac{n m}{2} - \frac{m n}{2} - 2 n (- 2 n)$

Step 3.1.7

Rewrite using the commutative property of multiplication.

$\frac{m^{2}}{16} - \frac{n m}{2} - \frac{m n}{2} - 2 \cdot - 2 n \cdot n$

Step 3.1.8

Multiply $n$ by $n$ by adding the exponents.

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Step 3.1.8.1

Move $n$ .

$\frac{m^{2}}{16} - \frac{n m}{2} - \frac{m n}{2} - 2 \cdot - 2 (n \cdot n)$

Step 3.1.8.2

Multiply $n$ by $n$ .

$\frac{m^{2}}{16} - \frac{n m}{2} - \frac{m n}{2} - 2 \cdot - 2 n^{2}$

Step 3.1.9

Multiply $- 2$ by $- 2$ .

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

Step 3.2

Subtract $\frac{m n}{2}$ from $- \frac{n m}{2}$ .

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Step 3.2.1

Reorder $n$ and $m$ .

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

Step 3.2.2

Subtract $\frac{m n}{2}$ from $- \frac{m n}{2}$ .

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

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$\frac{m^{2}}{16} + 2 (- 1) \frac{m n}{2} + 4 n^{2}$

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

$\frac{m^{2}}{16} - 1 (m n) + 4 n^{2}$

Step 4.2

Rewrite $- 1 (m n)$ as $- (m n)$ .

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