Expand Using Pascal's Triangle (a+b)^8

${(a + b)}^{8}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

$1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$

$1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$

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 ${(a + b)}^{8}$ , $n = 8$ so the coefficients of the expansion will correspond with line $9$ .

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 $1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$ .

$1 a^{8} b^{0} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + 1 a^{0} b^{8}$

Step 3

Substitute the actual values of $a$ $a$ and $b$ $b$ into the expression.

$1 {(a)}^{8} {(b)}^{0} + 8 {(a)}^{7} {(b)}^{1} + 28 {(a)}^{6} {(b)}^{2} + 56 {(a)}^{5} {(b)}^{3} + 70 {(a)}^{4} {(b)}^{4} + 56 {(a)}^{3} {(b)}^{5} + 28 {(a)}^{2} {(b)}^{6} + 8 {(a)}^{1} {(b)}^{7} + 1 {(a)}^{0} {(b)}^{8}$

Step 4

Simplify each term.

Multiply ${(a)}^{8}$ by $1$ .

${(a)}^{8} {(b)}^{0} + 8 {(a)}^{7} {(b)}^{1} + 28 {(a)}^{6} {(b)}^{2} + 56 {(a)}^{5} {(b)}^{3} + 70 {(a)}^{4} {(b)}^{4} + 56 {(a)}^{3} {(b)}^{5} + 28 {(a)}^{2} {(b)}^{6} + 8 {(a)}^{1} {(b)}^{7} + 1 {(a)}^{0} {(b)}^{8}$

Anything raised to $0$ is $1$ .

$a^{8} \cdot 1 + 8 {(a)}^{7} {(b)}^{1} + 28 {(a)}^{6} {(b)}^{2} + 56 {(a)}^{5} {(b)}^{3} + 70 {(a)}^{4} {(b)}^{4} + 56 {(a)}^{3} {(b)}^{5} + 28 {(a)}^{2} {(b)}^{6} + 8 {(a)}^{1} {(b)}^{7} + 1 {(a)}^{0} {(b)}^{8}$

Multiply $a^{8}$ by $1$ .

$a^{8} + 8 {(a)}^{7} {(b)}^{1} + 28 {(a)}^{6} {(b)}^{2} + 56 {(a)}^{5} {(b)}^{3} + 70 {(a)}^{4} {(b)}^{4} + 56 {(a)}^{3} {(b)}^{5} + 28 {(a)}^{2} {(b)}^{6} + 8 {(a)}^{1} {(b)}^{7} + 1 {(a)}^{0} {(b)}^{8}$

Simplify.

$a^{8} + 8 a^{7} b + 28 {(a)}^{6} {(b)}^{2} + 56 {(a)}^{5} {(b)}^{3} + 70 {(a)}^{4} {(b)}^{4} + 56 {(a)}^{3} {(b)}^{5} + 28 {(a)}^{2} {(b)}^{6} + 8 {(a)}^{1} {(b)}^{7} + 1 {(a)}^{0} {(b)}^{8}$

Simplify.

$a^{8} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a {(b)}^{7} + 1 {(a)}^{0} {(b)}^{8}$

Multiply ${(a)}^{0}$ by $1$ .

$a^{8} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + {(a)}^{0} {(b)}^{8}$

Anything raised to $0$ is $1$ .

$a^{8} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + 1 {(b)}^{8}$

Multiply ${(b)}^{8}$ by $1$ .

$a^{8} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + b^{8}$

Expand Using Pascal's Triangle (a+b)^8

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