Algebra Examples

${(a + b)}^{6}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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

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 ${(a + b)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

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

$1 {(a)}^{6} {(b)}^{0} + 6 {(a)}^{5} {(b)}^{1} + 15 {(a)}^{4} {(b)}^{2} + 20 {(a)}^{3} {(b)}^{3} + 15 {(a)}^{2} {(b)}^{4} + 6 {(a)}^{1} {(b)}^{5} + 1 {(a)}^{0} {(b)}^{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(a)}^{6}$ by $1$ .

${(a)}^{6} {(b)}^{0} + 6 {(a)}^{5} {(b)}^{1} + 15 {(a)}^{4} {(b)}^{2} + 20 {(a)}^{3} {(b)}^{3} + 15 {(a)}^{2} {(b)}^{4} + 6 {(a)}^{1} {(b)}^{5} + 1 {(a)}^{0} {(b)}^{6}$

Step 4.2

Anything raised to $0$ is $1$ .

$a^{6} \cdot 1 + 6 {(a)}^{5} {(b)}^{1} + 15 {(a)}^{4} {(b)}^{2} + 20 {(a)}^{3} {(b)}^{3} + 15 {(a)}^{2} {(b)}^{4} + 6 {(a)}^{1} {(b)}^{5} + 1 {(a)}^{0} {(b)}^{6}$

Step 4.3

Multiply $a^{6}$ by $1$ .

$a^{6} + 6 {(a)}^{5} {(b)}^{1} + 15 {(a)}^{4} {(b)}^{2} + 20 {(a)}^{3} {(b)}^{3} + 15 {(a)}^{2} {(b)}^{4} + 6 {(a)}^{1} {(b)}^{5} + 1 {(a)}^{0} {(b)}^{6}$

Step 4.4

Simplify.

$a^{6} + 6 a^{5} b + 15 {(a)}^{4} {(b)}^{2} + 20 {(a)}^{3} {(b)}^{3} + 15 {(a)}^{2} {(b)}^{4} + 6 {(a)}^{1} {(b)}^{5} + 1 {(a)}^{0} {(b)}^{6}$

Step 4.5

Simplify.

$a^{6} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a {(b)}^{5} + 1 {(a)}^{0} {(b)}^{6}$

Step 4.6

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

$a^{6} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + {(a)}^{0} {(b)}^{6}$

Step 4.7

Anything raised to $0$ is $1$ .

$a^{6} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 {(b)}^{6}$

Step 4.8

Multiply ${(b)}^{6}$ by $1$ .

$a^{6} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + b^{6}$