Algebra Examples

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $0$ .

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

Step 4.3

Anything raised to $0$ is $1$ .

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

Step 4.4

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

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

Step 4.5

Simplify.

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Multiply $2$ by $1$ .

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

Step 4.7

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

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

Step 4.8

Anything raised to $0$ is $1$ .

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

Step 4.9

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

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

Step 4.10

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

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Step 4.10.1

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

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

Step 4.10.2

Multiply $2$ by $2$ .

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