Expand Using Pascal's Triangle (x+5)^4

${(x + 5)}^{4}$

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 ${(x + 5)}^{4}$ , $n = 4$ 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 $a$ $x$ and $b$ $5$ into the expression.

$1 {(x)}^{4} {(5)}^{0} + 4 {(x)}^{3} {(5)}^{1} + 6 {(x)}^{2} {(5)}^{2} + 4 {(x)}^{1} {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

Step 4

Simplify each term.

Multiply ${(x)}^{4}$ by $1$ .

${(x)}^{4} {(5)}^{0} + 4 {(x)}^{3} {(5)}^{1} + 6 {(x)}^{2} {(5)}^{2} + 4 {(x)}^{1} {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

Anything raised to $0$ is $1$ .

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

Multiply $x^{4}$ by $1$ .

$x^{4} + 4 {(x)}^{3} {(5)}^{1} + 6 {(x)}^{2} {(5)}^{2} + 4 {(x)}^{1} {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

Evaluate the exponent.

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

Multiply $5$ by $4$ .

$x^{4} + 20 x^{3} + 6 {(x)}^{2} {(5)}^{2} + 4 {(x)}^{1} {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

Raise $5$ to the power of $2$ .

$x^{4} + 20 x^{3} + 6 x^{2} \cdot 25 + 4 {(x)}^{1} {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

Multiply $25$ by $6$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 4 {(x)}^{1} {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

Simplify.

$x^{4} + 20 x^{3} + 150 x^{2} + 4 x {(5)}^{3} + 1 {(x)}^{0} {(5)}^{4}$

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

$x^{4} + 20 x^{3} + 150 x^{2} + 4 x \cdot 125 + 1 {(x)}^{0} {(5)}^{4}$

Multiply $125$ by $4$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 1 {(x)}^{0} {(5)}^{4}$

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

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + {(x)}^{0} {(5)}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 1 {(5)}^{4}$

Multiply ${(5)}^{4}$ by $1$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + {(5)}^{4}$

Raise $5$ to the power of $4$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625$

Expand Using Pascal's Triangle (x+5)^4

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