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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Multiply the exponents in ${(x^{3})}^{4}$ .

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

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

Multiply $3$ by $4$ .

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

Apply the product rule to $- y^{2}$ .

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{12}$ by $1$ .

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

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

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

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

Multiply $2$ by $0$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{12}$ by $1$ .

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

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

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

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

Multiply $3$ by $3$ .

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $4$ by $- 1$ .

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

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

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

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

Multiply $3$ by $2$ .

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

Apply the product rule to $- y^{2}$ .

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

Rewrite using the commutative property of multiplication.

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

Raise $- 1$ to the power of $2$ .

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

Multiply $6$ by $1$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Multiply the exponents in ${(x^{3})}^{1}$ .

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

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

Multiply $3$ by $1$ .

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

Apply the product rule to $- y^{2}$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $4$ by $- 1$ .

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply the exponents in ${(x^{3})}^{0}$ .

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

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

Multiply $3$ by $0$ .

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

Anything raised to $0$ is $1$ .

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

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

$x^{12} - 4 x^{9} y^{2} + 6 x^{6} y^{4} - 4 x^{3} y^{6} + {(- y^{2})}^{4}$

Apply the product rule to $- y^{2}$ .

$x^{12} - 4 x^{9} y^{2} + 6 x^{6} y^{4} - 4 x^{3} y^{6} + {(- 1)}^{4} {(y^{2})}^{4}$

Raise $- 1$ to the power of $4$ .

$x^{12} - 4 x^{9} y^{2} + 6 x^{6} y^{4} - 4 x^{3} y^{6} + 1 {(y^{2})}^{4}$

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

$x^{12} - 4 x^{9} y^{2} + 6 x^{6} y^{4} - 4 x^{3} y^{6} + {(y^{2})}^{4}$

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

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

$x^{12} - 4 x^{9} y^{2} + 6 x^{6} y^{4} - 4 x^{3} y^{6} + y^{2 \cdot 4}$

Multiply $2$ by $4$ .

$x^{12} - 4 x^{9} y^{2} + 6 x^{6} y^{4} - 4 x^{3} y^{6} + y^{8}$

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

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