Simplify (1-2n)^3-7n(n^2-2)

${(1 - 2 n)}^{3} - 7 n (n^{2} - 2)$

Simplify each term.

Use the Binomial Theorem.

$1^{3} + 3 \cdot 1^{2} (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Simplify each term.

One to any power is one.

$1 + 3 \cdot 1^{2} (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

One to any power is one.

$1 + 3 \cdot 1 (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Multiply $3$ by $1$ .

$1 + 3 (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Multiply $- 2$ by $3$ .

$1 - 6 n + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Multiply $3$ by $1$ .

$1 - 6 n + 3 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Apply the product rule to $- 2 n$ .

$1 - 6 n + 3 ({(- 2)}^{2} n^{2}) + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Raise $- 2$ to the power of $2$ .

$1 - 6 n + 3 (4 n^{2}) + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Multiply $4$ by $3$ .

$1 - 6 n + 12 n^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Apply the product rule to $- 2 n$ .

$1 - 6 n + 12 n^{2} + {(- 2)}^{3} n^{3} - 7 n (n^{2} - 2)$

Raise $- 2$ to the power of $3$ .

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)$

Apply the distributive property.

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n \cdot n^{2} - 7 n \cdot - 2$

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

Move $n^{2}$ .

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 (n^{2} n) - 7 n \cdot - 2$

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

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

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 (n^{2} n^{1}) - 7 n \cdot - 2$

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

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n^{2 + 1} - 7 n \cdot - 2$

Add $2$ and $1$ .

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n^{3} - 7 n \cdot - 2$

Multiply $- 2$ by $- 7$ .

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n^{3} + 14 n$

Simplify by adding terms.

Add $- 6 n$ and $14 n$ .

$1 + 12 n^{2} - 8 n^{3} - 7 n^{3} + 8 n$

Subtract $7 n^{3}$ from $- 8 n^{3}$ .

$1 + 12 n^{2} - 15 n^{3} + 8 n$

Simplify the expression.

Move $1$ .

$12 n^{2} - 15 n^{3} + 8 n + 1$

Reorder $12 n^{2}$ and $- 15 n^{3}$ .

$- 15 n^{3} + 12 n^{2} + 8 n + 1$

Simplify (1-2n)^3-7n(n^2-2)

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