Expand Using Pascal's Triangle (2p^4-3t)^4

${(2 p^{4} - 3 t)}^{4}$

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 ${(2 p^{4} - 3 t)}^{4}$ , $n = 4$ 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 $a$ $2 p^{4}$ and $b$ $- 3 t$ into the expression.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $2 p^{4}$ .

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

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

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

Multiply the exponents in ${(p^{4})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 p^{4 \cdot 4} {(- 3 t)}^{0} + 4 {(2 p^{4})}^{3} {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $4$ by $4$ .

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

Apply the product rule to $- 3 t$ .

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

Rewrite using the commutative property of multiplication.

$16 \cdot {(- 3)}^{0} p^{16} t^{0} + 4 {(2 p^{4})}^{3} {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Anything raised to $0$ is $1$ .

$16 \cdot 1 p^{16} t^{0} + 4 {(2 p^{4})}^{3} {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $16$ by $1$ .

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

Anything raised to $0$ is $1$ .

$16 p^{16} \cdot 1 + 4 {(2 p^{4})}^{3} {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $16$ by $1$ .

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

Apply the product rule to $2 p^{4}$ .

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

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

$16 p^{16} + 4 (8 {(p^{4})}^{3}) {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply the exponents in ${(p^{4})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 p^{16} + 4 (8 p^{4 \cdot 3}) {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $4$ by $3$ .

$16 p^{16} + 4 (8 p^{12}) {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $8$ by $4$ .

$16 p^{16} + 32 p^{12} {(- 3 t)}^{1} + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Simplify.

$16 p^{16} + 32 p^{12} (- 3 t) + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

$16 p^{16} + 32 \cdot - 3 p^{12} t + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $32$ by $- 3$ .

$16 p^{16} - 96 p^{12} t + 6 {(2 p^{4})}^{2} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Apply the product rule to $2 p^{4}$ .

$16 p^{16} - 96 p^{12} t + 6 (2^{2} {(p^{4})}^{2}) {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

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

$16 p^{16} - 96 p^{12} t + 6 (4 {(p^{4})}^{2}) {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply the exponents in ${(p^{4})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 p^{16} - 96 p^{12} t + 6 (4 p^{4 \cdot 2}) {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $4$ by $2$ .

$16 p^{16} - 96 p^{12} t + 6 (4 p^{8}) {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $4$ by $6$ .

$16 p^{16} - 96 p^{12} t + 24 p^{8} {(- 3 t)}^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$16 p^{16} - 96 p^{12} t + 24 p^{8} ({(- 3)}^{2} t^{2}) + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

$16 p^{16} - 96 p^{12} t + 24 \cdot {(- 3)}^{2} p^{8} t^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

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

$16 p^{16} - 96 p^{12} t + 24 \cdot 9 p^{8} t^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $24$ by $9$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} + 4 {(2 p^{4})}^{1} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Simplify.

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} + 4 (2 p^{4}) {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $2$ by $4$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} + 8 p^{4} {(- 3 t)}^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} + 8 p^{4} ({(- 3)}^{3} t^{3}) + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} + 8 \cdot {(- 3)}^{3} p^{4} t^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

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

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} + 8 \cdot - 27 p^{4} t^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply $8$ by $- 27$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + 1 {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Multiply ${(2 p^{4})}^{0}$ by $1$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + {(2 p^{4})}^{0} {(- 3 t)}^{4}$

Apply the product rule to $2 p^{4}$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + 2^{0} {(p^{4})}^{0} {(- 3 t)}^{4}$

Anything raised to $0$ is $1$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + 1 {(p^{4})}^{0} {(- 3 t)}^{4}$

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

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + {(p^{4})}^{0} {(- 3 t)}^{4}$

Multiply the exponents in ${(p^{4})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + p^{4 \cdot 0} {(- 3 t)}^{4}$

Multiply $4$ by $0$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + p^{0} {(- 3 t)}^{4}$

Anything raised to $0$ is $1$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + 1 {(- 3 t)}^{4}$

Multiply ${(- 3 t)}^{4}$ by $1$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + {(- 3)}^{4} t^{4}$

Raise $- 3$ to the power of $4$ .

$16 p^{16} - 96 p^{12} t + 216 p^{8} t^{2} - 216 p^{4} t^{3} + 81 t^{4}$

Expand Using Pascal's Triangle (2p^4-3t)^4

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