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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

Multiply ${(x^{2})}^{3}$ by $1$ .

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{6}$ by $1$ .

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

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

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

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

Multiply $2$ by $0$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{6}$ by $1$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $- 1$ .

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

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

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

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

Multiply $2$ by $1$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $3$ by $1$ .

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

Multiply $2$ by $0$ .

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

Anything raised to $0$ is $1$ .

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

Multiply ${(- y^{2})}^{3}$ by $1$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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