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

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

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

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

Step 4

Simplify each term.

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

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

Apply the product rule to $3 x$ .

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

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

$243 x^{5} {(5)}^{0} + 5 {(3 x)}^{4} {(5)}^{1} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Anything raised to $0$ is $1$ .

$243 x^{5} \cdot 1 + 5 {(3 x)}^{4} {(5)}^{1} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $243$ by $1$ .

$243 x^{5} + 5 {(3 x)}^{4} {(5)}^{1} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Apply the product rule to $3 x$ .

$243 x^{5} + 5 (3^{4} x^{4}) {(5)}^{1} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 5 (81 x^{4}) {(5)}^{1} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $81$ by $5$ .

$243 x^{5} + 405 x^{4} {(5)}^{1} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Evaluate the exponent.

$243 x^{5} + 405 x^{4} \cdot 5 + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $5$ by $405$ .

$243 x^{5} + 2025 x^{4} + 10 {(3 x)}^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Apply the product rule to $3 x$ .

$243 x^{5} + 2025 x^{4} + 10 (3^{3} x^{3}) {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 10 (27 x^{3}) {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $27$ by $10$ .

$243 x^{5} + 2025 x^{4} + 270 x^{3} {(5)}^{2} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 270 x^{3} \cdot 25 + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $25$ by $270$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 10 {(3 x)}^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Apply the product rule to $3 x$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 10 (3^{2} x^{2}) {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 10 (9 x^{2}) {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $9$ by $10$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 90 x^{2} {(5)}^{3} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 90 x^{2} \cdot 125 + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $125$ by $90$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 5 {(3 x)}^{1} {(5)}^{4} + 1 {(3 x)}^{0} {(5)}^{5}$

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

Move ${(5)}^{4}$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + {(5)}^{4} \cdot 5 {(3 x)}^{1} + 1 {(3 x)}^{0} {(5)}^{5}$

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

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + {(5)}^{4} \cdot 5^{1} {(3 x)}^{1} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 5^{4 + 1} {(3 x)}^{1} + 1 {(3 x)}^{0} {(5)}^{5}$

Add $4$ and $1$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 5^{5} {(3 x)}^{1} + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 5^{5} (3 x) + 1 {(3 x)}^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 3125 (3 x) + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply $3$ by $3125$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 9375 x + 1 {(3 x)}^{0} {(5)}^{5}$

Multiply ${(3 x)}^{0}$ by $1$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 9375 x + {(3 x)}^{0} {(5)}^{5}$

Apply the product rule to $3 x$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 9375 x + 3^{0} x^{0} {(5)}^{5}$

Anything raised to $0$ is $1$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 9375 x + 1 x^{0} {(5)}^{5}$

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

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 9375 x + x^{0} {(5)}^{5}$

Anything raised to $0$ is $1$ .

$243 x^{5} + 2025 x^{4} + 6750 x^{3} + 11250 x^{2} + 9375 x + 1 {(5)}^{5}$

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

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

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

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

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

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