Basic Math Examples

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

Step 1

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

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

Step 2

Expand $(2 n^{2} - n - 1) (2 n^{2} - n - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

Multiply $n^{2}$ by $n^{2}$ by adding the exponents.

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Step 3.1.2.1

Move $n^{2}$ .

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Add $2$ and $2$ .

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

Step 3.1.3

Multiply $2$ by $2$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

Multiply $n^{2}$ by $n$ by adding the exponents.

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Step 3.1.5.1

Move $n$ .

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

Step 3.1.5.2

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

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Step 3.1.5.2.1

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

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

Step 3.1.5.2.2

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

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

Step 3.1.5.3

Add $1$ and $2$ .

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

Step 3.1.6

Multiply $2$ by $- 1$ .

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

Step 3.1.7

Multiply $- 1$ by $2$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - n (2 n^{2}) - n (- n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.8

Rewrite using the commutative property of multiplication.

$4 n^{4} - 2 n^{3} - 2 n^{2} - 1 \cdot 2 n \cdot n^{2} - n (- n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.9

Multiply $n$ by $n^{2}$ by adding the exponents.

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Step 3.1.9.1

Move $n^{2}$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 1 \cdot 2 (n^{2} n) - n (- n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.9.2

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

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Step 3.1.9.2.1

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

$4 n^{4} - 2 n^{3} - 2 n^{2} - 1 \cdot 2 (n^{2} n^{1}) - n (- n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.9.2.2

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

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

Step 3.1.9.3

Add $2$ and $1$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 1 \cdot 2 n^{3} - n (- n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.10

Multiply $- 1$ by $2$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} - n (- n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.11

Rewrite using the commutative property of multiplication.

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} - 1 \cdot - 1 n \cdot n - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.12

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

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Step 3.1.12.1

Move $n$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} - 1 \cdot - 1 (n \cdot n) - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.12.2

Multiply $n$ by $n$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} - 1 \cdot - 1 n^{2} - n \cdot - 1 - 1 (2 n^{2}) - 1 (- n) - 1 \cdot - 1$

Step 3.1.13

Multiply $- 1$ by $- 1$ .

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

Step 3.1.14

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

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

Step 3.1.15

Multiply $- n \cdot - 1$ .

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Step 3.1.15.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.15.2

Multiply $n$ by $1$ .

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

Step 3.1.16

Multiply $2$ by $- 1$ .

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

Step 3.1.17

Multiply $- 1 (- n)$ .

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Step 3.1.17.1

Multiply $- 1$ by $- 1$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} + n^{2} + n - 2 n^{2} + 1 n - 1 \cdot - 1$

Step 3.1.17.2

Multiply $n$ by $1$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} + n^{2} + n - 2 n^{2} + n - 1 \cdot - 1$

Step 3.1.18

Multiply $- 1$ by $- 1$ .

$4 n^{4} - 2 n^{3} - 2 n^{2} - 2 n^{3} + n^{2} + n - 2 n^{2} + n + 1$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $2 n^{3}$ from $- 2 n^{3}$ .

$4 n^{4} - 4 n^{3} - 2 n^{2} + n^{2} + n - 2 n^{2} + n + 1$

Step 3.2.2

Add $- 2 n^{2}$ and $n^{2}$ .

$4 n^{4} - 4 n^{3} - n^{2} + n - 2 n^{2} + n + 1$

Step 3.2.3

Subtract $2 n^{2}$ from $- n^{2}$ .

$4 n^{4} - 4 n^{3} - 3 n^{2} + n + n + 1$

Step 3.2.4

Add $n$ and $n$ .

$4 n^{4} - 4 n^{3} - 3 n^{2} + 2 n + 1$