Basic Math Examples

${(m - n + 1)}^{2}$

Step 1

Rewrite ${(m - n + 1)}^{2}$ as $(m - n + 1) (m - n + 1)$ .

$(m - n + 1) (m - n + 1)$

Step 2

Expand $(m - n + 1) (m - n + 1)$ by multiplying each term in the first expression by each term in the second expression.

$m \cdot m + m (- n) + m \cdot 1 - n m - n (- n) - n \cdot 1 + 1 m + 1 (- n) + 1 \cdot 1$

Step 3

Simplify each term.

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Step 3.1

Multiply $m$ by $m$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Multiply $m$ by $1$ .

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

Step 3.4

Rewrite using the commutative property of multiplication.

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

Step 3.5

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

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Step 3.5.1

Move $n$ .

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

Step 3.5.2

Multiply $n$ by $n$ .

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

Step 3.6

Multiply $- 1$ by $- 1$ .

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

Step 3.7

Multiply $n^{2}$ by $1$ .

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

Step 3.8

Multiply $- 1$ by $1$ .

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

Step 3.9

Multiply $m$ by $1$ .

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

Step 3.10

Multiply $- n$ by $1$ .

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

Step 3.11

Multiply $1$ by $1$ .

$m^{2} - m n + m - n m + n^{2} - n + m - n + 1$

Step 4

Subtract $n m$ from $- m n$ .

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Step 4.1

Move $n$ .

$m^{2} - m n - 1 m n + m + n^{2} - n + m - n + 1$

Step 4.2

Subtract $m n$ from $- m n$ .

$m^{2} - 2 m n + m + n^{2} - n + m - n + 1$

Step 5

Add $m$ and $m$ .

$m^{2} - 2 m n + 2 m + n^{2} - n - n + 1$

Step 6

Subtract $n$ from $- n$ .

$m^{2} - 2 m n + 2 m + n^{2} - 2 n + 1$

Step 7

Move $2 m$ .

$m^{2} - 2 m n + n^{2} + 2 m - 2 n + 1$