Expand Using the Binomial Theorem (x^2+3y)^4

${(x^{2} + 3 y)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(x^{2})}^{4 - k} \cdot {(3 y)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

Multiply ${(x^{2})}^{4}$ by $1$ .

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

Multiply the exponents in ${(x^{2})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $4$ .

$x^{8} \cdot {(3 y)}^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$x^{8} \cdot (3^{0} y^{0}) + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$3^{0} x^{8} y^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$1 x^{8} y^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $x^{8}$ by $1$ .

$x^{8} y^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$x^{8} \cdot 1 + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $x^{8}$ by $1$ .

$x^{8} + 4 \cdot {(x^{2})}^{3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} + 4 \cdot x^{2 \cdot 3} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $2$ by $3$ .

$x^{8} + 4 \cdot x^{6} \cdot {(3 y)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Simplify.

$x^{8} + 4 x^{6} \cdot (3 y) + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{8} + 4 \cdot 3 x^{6} y + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $4$ by $3$ .

$x^{8} + 12 x^{6} y + 6 \cdot {(x^{2})}^{2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} + 12 x^{6} y + 6 \cdot x^{2 \cdot 2} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $2$ by $2$ .

$x^{8} + 12 x^{6} y + 6 \cdot x^{4} \cdot {(3 y)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$x^{8} + 12 x^{6} y + 6 x^{4} \cdot (3^{2} y^{2}) + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{8} + 12 x^{6} y + 6 \cdot 3^{2} x^{4} y^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

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

$x^{8} + 12 x^{6} y + 6 \cdot 9 x^{4} y^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $6$ by $9$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply the exponents in ${(x^{2})}^{1}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 4 \cdot x^{2 \cdot 1} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $2$ by $1$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 4 \cdot x^{2} \cdot {(3 y)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 4 x^{2} \cdot (3^{3} y^{3}) + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 4 \cdot 3^{3} x^{2} y^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

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

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 4 \cdot 27 x^{2} y^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply $4$ by $27$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply ${(x^{2})}^{0}$ by $1$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + {(x^{2})}^{0} \cdot {(3 y)}^{4}$

Multiply the exponents in ${(x^{2})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + x^{2 \cdot 0} \cdot {(3 y)}^{4}$

Multiply $2$ by $0$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + x^{0} \cdot {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + 1 \cdot {(3 y)}^{4}$

Multiply ${(3 y)}^{4}$ by $1$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + 3^{4} y^{4}$

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

$x^{8} + 12 x^{6} y + 54 x^{4} y^{2} + 108 x^{2} y^{3} + 81 y^{4}$

Expand Using the Binomial Theorem (x^2+3y)^4

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