Expand Using Pascal's Triangle (4t-2)^4

${(4 t - 2)}^{4}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(4 t - 2)}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Step 3

Substitute the actual values of $a$ $4 t$ and $b$ $- 2$ into the expression.

$1 {(4 t)}^{4} {(- 2)}^{0} + 4 {(4 t)}^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Step 4

Simplify each term.

Multiply ${(4 t)}^{4}$ by $1$ .

${(4 t)}^{4} {(- 2)}^{0} + 4 {(4 t)}^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Apply the product rule to $4 t$ .

$4^{4} t^{4} {(- 2)}^{0} + 4 {(4 t)}^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Raise $4$ to the power of $4$ .

$256 t^{4} {(- 2)}^{0} + 4 {(4 t)}^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Anything raised to $0$ is $1$ .

$256 t^{4} \cdot 1 + 4 {(4 t)}^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $256$ by $1$ .

$256 t^{4} + 4 {(4 t)}^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Apply the product rule to $4 t$ .

$256 t^{4} + 4 (4^{3} t^{3}) {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $4$ by $4^{3}$ by adding the exponents.

Move $4^{3}$ .

$256 t^{4} + 4^{3} \cdot 4 t^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $4^{3}$ by $4$ .

Raise $4$ to the power of $1$ .

$256 t^{4} + 4^{3} \cdot 4^{1} t^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

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

$256 t^{4} + 4^{3 + 1} t^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Add $3$ and $1$ .

$256 t^{4} + 4^{4} t^{3} {(- 2)}^{1} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Simplify $4^{4} t^{3} {(- 2)}^{1}$ .

$256 t^{4} + 4^{4} t^{3} \cdot - 2 + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Raise $4$ to the power of $4$ .

$256 t^{4} + 256 t^{3} \cdot - 2 + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $- 2$ by $256$ .

$256 t^{4} - 512 t^{3} + 6 {(4 t)}^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Apply the product rule to $4 t$ .

$256 t^{4} - 512 t^{3} + 6 (4^{2} t^{2}) {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Raise $4$ to the power of $2$ .

$256 t^{4} - 512 t^{3} + 6 (16 t^{2}) {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $16$ by $6$ .

$256 t^{4} - 512 t^{3} + 96 t^{2} {(- 2)}^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

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

$256 t^{4} - 512 t^{3} + 96 t^{2} \cdot 4 + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $4$ by $96$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} + 4 {(4 t)}^{1} {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Simplify.

$256 t^{4} - 512 t^{3} + 384 t^{2} + 4 (4 t) {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $4$ by $4$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} + 16 t {(- 2)}^{3} + 1 {(4 t)}^{0} {(- 2)}^{4}$

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

$256 t^{4} - 512 t^{3} + 384 t^{2} + 16 t \cdot - 8 + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply $- 8$ by $16$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + 1 {(4 t)}^{0} {(- 2)}^{4}$

Multiply ${(4 t)}^{0}$ by $1$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + {(4 t)}^{0} {(- 2)}^{4}$

Apply the product rule to $4 t$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + 4^{0} t^{0} {(- 2)}^{4}$

Anything raised to $0$ is $1$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + 1 t^{0} {(- 2)}^{4}$

Multiply $t^{0}$ by $1$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + t^{0} {(- 2)}^{4}$

Anything raised to $0$ is $1$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + 1 {(- 2)}^{4}$

Multiply ${(- 2)}^{4}$ by $1$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + {(- 2)}^{4}$

Raise $- 2$ to the power of $4$ .

$256 t^{4} - 512 t^{3} + 384 t^{2} - 128 t + 16$

Expand Using Pascal's Triangle (4t-2)^4

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