Expand Using the Binomial Theorem (3x+3y)^3

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(3 x)}^{3 - k} \cdot {(3 y)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} \cdot {(3 x)}^{3 - 0} \cdot {(3 y)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(3 x)}^{3 - 1} \cdot {(3 y)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(3 x)}^{3 - 2} \cdot {(3 y)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(3 x)}^{3 - 3} \cdot {(3 y)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3 x)}^{3} \cdot {(3 y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Step 4

Simplify each term.

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

${(3 x)}^{3} \cdot {(3 y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Apply the product rule to $3 x$ .

$3^{3} x^{3} \cdot {(3 y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} \cdot {(3 y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Apply the product rule to $3 y$ .

$27 x^{3} \cdot (3^{0} y^{0}) + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Rewrite using the commutative property of multiplication.

$27 \cdot 3^{0} x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Anything raised to $0$ is $1$ .

$27 \cdot 1 x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $27$ by $1$ .

$27 x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Anything raised to $0$ is $1$ .

$27 x^{3} \cdot 1 + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $27$ by $1$ .

$27 x^{3} + 3 \cdot {(3 x)}^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Apply the product rule to $3 x$ .

$27 x^{3} + 3 \cdot (3^{2} x^{2}) \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $3$ by $3^{2}$ by adding the exponents.

Move $3^{2}$ .

$27 x^{3} + 3^{2} \cdot 3 \cdot x^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

$27 x^{3} + 3^{2} \cdot 3^{1} \cdot x^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} + 3^{2 + 1} \cdot x^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Add $2$ and $1$ .

$27 x^{3} + 3^{3} \cdot x^{2} \cdot {(3 y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Simplify $3^{3} \cdot x^{2} \cdot {(3 y)}^{1}$ .

$27 x^{3} + 3^{3} \cdot x^{2} \cdot (3 y) + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Rewrite using the commutative property of multiplication.

$27 x^{3} + 3^{3} \cdot 3 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $3^{3}$ by $3$ by adding the exponents.

Multiply $3^{3}$ by $3$ .

Raise $3$ to the power of $1$ .

$27 x^{3} + 3^{3} \cdot 3^{1} x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} + 3^{3 + 1} x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Add $3$ and $1$ .

$27 x^{3} + 3^{4} x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} + 81 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Simplify.

$27 x^{3} + 81 x^{2} y + 3 \cdot (3 x) \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $3$ by $3$ .

$27 x^{3} + 81 x^{2} y + 9 x \cdot {(3 y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Apply the product rule to $3 y$ .

$27 x^{3} + 81 x^{2} y + 9 x \cdot (3^{2} y^{2}) + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Rewrite using the commutative property of multiplication.

$27 x^{3} + 81 x^{2} y + 9 \cdot 3^{2} x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} + 81 x^{2} y + 9 \cdot 9 x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

Multiply $9$ by $9$ .

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + {(3 x)}^{0} \cdot {(3 y)}^{3}$

Apply the product rule to $3 x$ .

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + 3^{0} x^{0} \cdot {(3 y)}^{3}$

Anything raised to $0$ is $1$ .

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + 1 x^{0} \cdot {(3 y)}^{3}$

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

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + x^{0} \cdot {(3 y)}^{3}$

Anything raised to $0$ is $1$ .

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + 1 \cdot {(3 y)}^{3}$

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

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + {(3 y)}^{3}$

Apply the product rule to $3 y$ .

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + 3^{3} y^{3}$

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

$27 x^{3} + 81 x^{2} y + 81 x y^{2} + 27 y^{3}$

Expand Using the Binomial Theorem (3x+3y)^3

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