Expand Using Pascal's Triangle (r+3)^5

${(r + 3)}^{5}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(r + 3)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $r$ and $b$ $3$ into the expression.

$1 {(r)}^{5} {(3)}^{0} + 5 {(r)}^{4} {(3)}^{1} + 10 {(r)}^{3} {(3)}^{2} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Step 4

Simplify each term.

Multiply ${(r)}^{5}$ by $1$ .

${(r)}^{5} {(3)}^{0} + 5 {(r)}^{4} {(3)}^{1} + 10 {(r)}^{3} {(3)}^{2} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Anything raised to $0$ is $1$ .

$r^{5} \cdot 1 + 5 {(r)}^{4} {(3)}^{1} + 10 {(r)}^{3} {(3)}^{2} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Multiply $r^{5}$ by $1$ .

$r^{5} + 5 {(r)}^{4} {(3)}^{1} + 10 {(r)}^{3} {(3)}^{2} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Evaluate the exponent.

$r^{5} + 5 r^{4} \cdot 3 + 10 {(r)}^{3} {(3)}^{2} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Multiply $3$ by $5$ .

$r^{5} + 15 r^{4} + 10 {(r)}^{3} {(3)}^{2} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

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

$r^{5} + 15 r^{4} + 10 r^{3} \cdot 9 + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Multiply $9$ by $10$ .

$r^{5} + 15 r^{4} + 90 r^{3} + 10 {(r)}^{2} {(3)}^{3} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

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

$r^{5} + 15 r^{4} + 90 r^{3} + 10 r^{2} \cdot 27 + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Multiply $27$ by $10$ .

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 5 {(r)}^{1} {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

Simplify.

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 5 r {(3)}^{4} + 1 {(r)}^{0} {(3)}^{5}$

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

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 5 r \cdot 81 + 1 {(r)}^{0} {(3)}^{5}$

Multiply $81$ by $5$ .

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 405 r + 1 {(r)}^{0} {(3)}^{5}$

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

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 405 r + {(r)}^{0} {(3)}^{5}$

Anything raised to $0$ is $1$ .

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 405 r + 1 {(3)}^{5}$

Multiply ${(3)}^{5}$ by $1$ .

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 405 r + {(3)}^{5}$

Raise $3$ to the power of $5$ .

$r^{5} + 15 r^{4} + 90 r^{3} + 270 r^{2} + 405 r + 243$

Expand Using Pascal's Triangle (r+3)^5

Related Questions

Support LCMGCF.com by disabling your adblocker.