Expand Using Pascal's Triangle (w+x)^7

${(w + x)}^{7}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 5 - 10 - 10 - 5 - 1$

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

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

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 ${(w + x)}^{7}$ , $n = 7$ so the coefficients of the expansion will correspond with line $8$ .

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

$1 a^{7} b^{0} + 7 a^{6} b + 21 a^{5} b^{2} + 35 a^{4} b^{3} + 35 a^{3} b^{4} + 21 a^{2} b^{5} + 7 a b^{6} + 1 a^{0} b^{7}$

Step 3

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

$1 {(w)}^{7} {(x)}^{0} + 7 {(w)}^{6} {(x)}^{1} + 21 {(w)}^{5} {(x)}^{2} + 35 {(w)}^{4} {(x)}^{3} + 35 {(w)}^{3} {(x)}^{4} + 21 {(w)}^{2} {(x)}^{5} + 7 {(w)}^{1} {(x)}^{6} + 1 {(w)}^{0} {(x)}^{7}$

Step 4

Simplify each term.

Multiply ${(w)}^{7}$ by $1$ .

${(w)}^{7} {(x)}^{0} + 7 {(w)}^{6} {(x)}^{1} + 21 {(w)}^{5} {(x)}^{2} + 35 {(w)}^{4} {(x)}^{3} + 35 {(w)}^{3} {(x)}^{4} + 21 {(w)}^{2} {(x)}^{5} + 7 {(w)}^{1} {(x)}^{6} + 1 {(w)}^{0} {(x)}^{7}$

Anything raised to $0$ is $1$ .

$w^{7} \cdot 1 + 7 {(w)}^{6} {(x)}^{1} + 21 {(w)}^{5} {(x)}^{2} + 35 {(w)}^{4} {(x)}^{3} + 35 {(w)}^{3} {(x)}^{4} + 21 {(w)}^{2} {(x)}^{5} + 7 {(w)}^{1} {(x)}^{6} + 1 {(w)}^{0} {(x)}^{7}$

Multiply $w^{7}$ by $1$ .

$w^{7} + 7 {(w)}^{6} {(x)}^{1} + 21 {(w)}^{5} {(x)}^{2} + 35 {(w)}^{4} {(x)}^{3} + 35 {(w)}^{3} {(x)}^{4} + 21 {(w)}^{2} {(x)}^{5} + 7 {(w)}^{1} {(x)}^{6} + 1 {(w)}^{0} {(x)}^{7}$

Simplify.

$w^{7} + 7 w^{6} x + 21 {(w)}^{5} {(x)}^{2} + 35 {(w)}^{4} {(x)}^{3} + 35 {(w)}^{3} {(x)}^{4} + 21 {(w)}^{2} {(x)}^{5} + 7 {(w)}^{1} {(x)}^{6} + 1 {(w)}^{0} {(x)}^{7}$

Simplify.

$w^{7} + 7 w^{6} x + 21 w^{5} x^{2} + 35 w^{4} x^{3} + 35 w^{3} x^{4} + 21 w^{2} x^{5} + 7 w {(x)}^{6} + 1 {(w)}^{0} {(x)}^{7}$

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

$w^{7} + 7 w^{6} x + 21 w^{5} x^{2} + 35 w^{4} x^{3} + 35 w^{3} x^{4} + 21 w^{2} x^{5} + 7 w x^{6} + {(w)}^{0} {(x)}^{7}$

Anything raised to $0$ is $1$ .

$w^{7} + 7 w^{6} x + 21 w^{5} x^{2} + 35 w^{4} x^{3} + 35 w^{3} x^{4} + 21 w^{2} x^{5} + 7 w x^{6} + 1 {(x)}^{7}$

Multiply ${(x)}^{7}$ by $1$ .

$w^{7} + 7 w^{6} x + 21 w^{5} x^{2} + 35 w^{4} x^{3} + 35 w^{3} x^{4} + 21 w^{2} x^{5} + 7 w x^{6} + x^{7}$

Expand Using Pascal's Triangle (w+x)^7

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