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

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

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 ${(5 x + 3)}^{5}$ , $n = 5$ 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 $a$ $5 x$ and $b$ $3$ into the expression.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $5 x$ .

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

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

$3125 x^{5} {(3)}^{0} + 5 {(5 x)}^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Anything raised to $0$ is $1$ .

$3125 x^{5} \cdot 1 + 5 {(5 x)}^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $3125$ by $1$ .

$3125 x^{5} + 5 {(5 x)}^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Apply the product rule to $5 x$ .

$3125 x^{5} + 5 (5^{4} x^{4}) {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $5$ by $5^{4}$ by adding the exponents.

Move $5^{4}$ .

$3125 x^{5} + 5^{4} \cdot 5 x^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $5^{4}$ by $5$ .

Raise $5$ to the power of $1$ .

$3125 x^{5} + 5^{4} \cdot 5^{1} x^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3125 x^{5} + 5^{4 + 1} x^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Add $4$ and $1$ .

$3125 x^{5} + 5^{5} x^{4} {(3)}^{1} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Simplify $5^{5} x^{4} {(3)}^{1}$ .

$3125 x^{5} + 5^{5} x^{4} \cdot 3 + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

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

$3125 x^{5} + 3125 x^{4} \cdot 3 + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $3$ by $3125$ .

$3125 x^{5} + 9375 x^{4} + 10 {(5 x)}^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Apply the product rule to $5 x$ .

$3125 x^{5} + 9375 x^{4} + 10 (5^{3} x^{3}) {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 10 (125 x^{3}) {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $125$ by $10$ .

$3125 x^{5} + 9375 x^{4} + 1250 x^{3} {(3)}^{2} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 1250 x^{3} \cdot 9 + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $9$ by $1250$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 10 {(5 x)}^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Apply the product rule to $5 x$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 10 (5^{2} x^{2}) {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 10 (25 x^{2}) {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $25$ by $10$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 250 x^{2} {(3)}^{3} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 250 x^{2} \cdot 27 + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $27$ by $250$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 5 {(5 x)}^{1} {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Simplify.

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 5 (5 x) {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $5$ by $5$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 25 x {(3)}^{4} + 1 {(5 x)}^{0} {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 25 x \cdot 81 + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply $81$ by $25$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + 1 {(5 x)}^{0} {(3)}^{5}$

Multiply ${(5 x)}^{0}$ by $1$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + {(5 x)}^{0} {(3)}^{5}$

Apply the product rule to $5 x$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + 5^{0} x^{0} {(3)}^{5}$

Anything raised to $0$ is $1$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + 1 x^{0} {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + x^{0} {(3)}^{5}$

Anything raised to $0$ is $1$ .

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + 1 {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + {(3)}^{5}$

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

$3125 x^{5} + 9375 x^{4} + 11250 x^{3} + 6750 x^{2} + 2025 x + 243$

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

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