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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $0$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{4}$ by $1$ .

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

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

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

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

Multiply $2$ by $1$ .

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

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

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

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

Multiply $2$ by $1$ .

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

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

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

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

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

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

Multiply $2$ by $0$ .

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

Anything raised to $0$ is $1$ .

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

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

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

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

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

$x^{4} + 2 x^{2} y^{2} + y^{2 \cdot 2}$

Multiply $2$ by $2$ .

$x^{4} + 2 x^{2} y^{2} + y^{4}$

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

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