Algebra Examples

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $2 x$ .

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

Step 4.3

Raise $2$ to the power of $4$ .

$16 x^{4} \cdot {(y^{2})}^{0} + 4 \cdot {(2 x)}^{3} \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.4

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

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Step 4.4.1

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

$16 x^{4} \cdot y^{2 \cdot 0} + 4 \cdot {(2 x)}^{3} \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.4.2

Multiply $2$ by $0$ .

$16 x^{4} \cdot y^{0} + 4 \cdot {(2 x)}^{3} \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.5

Anything raised to $0$ is $1$ .

$16 x^{4} \cdot 1 + 4 \cdot {(2 x)}^{3} \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.6

Multiply $16$ by $1$ .

$16 x^{4} + 4 \cdot {(2 x)}^{3} \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.7

Apply the product rule to $2 x$ .

$16 x^{4} + 4 \cdot (2^{3} x^{3}) \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.8

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

$16 x^{4} + 4 \cdot (8 x^{3}) \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.9

Multiply $8$ by $4$ .

$16 x^{4} + 32 x^{3} \cdot {(y^{2})}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.10

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

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Step 4.10.1

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

$16 x^{4} + 32 x^{3} \cdot y^{2 \cdot 1} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.10.2

Multiply $2$ by $1$ .

$16 x^{4} + 32 x^{3} \cdot y^{2} + 6 \cdot {(2 x)}^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.11

Apply the product rule to $2 x$ .

$16 x^{4} + 32 x^{3} y^{2} + 6 \cdot (2^{2} x^{2}) \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.12

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

$16 x^{4} + 32 x^{3} y^{2} + 6 \cdot (4 x^{2}) \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.13

Multiply $4$ by $6$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} \cdot {(y^{2})}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.14

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

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Step 4.14.1

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

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} \cdot y^{2 \cdot 2} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.14.2

Multiply $2$ by $2$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} \cdot y^{4} + 4 \cdot {(2 x)}^{1} \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.15

Simplify.

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 4 \cdot (2 x) \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.16

Multiply $2$ by $4$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x \cdot {(y^{2})}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.17

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

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Step 4.17.1

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

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x \cdot y^{2 \cdot 3} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.17.2

Multiply $2$ by $3$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x \cdot y^{6} + 1 \cdot {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.18

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

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + {(2 x)}^{0} \cdot {(y^{2})}^{4}$

Step 4.19

Apply the product rule to $2 x$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + 2^{0} x^{0} \cdot {(y^{2})}^{4}$

Step 4.20

Anything raised to $0$ is $1$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + 1 x^{0} \cdot {(y^{2})}^{4}$

Step 4.21

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

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + x^{0} \cdot {(y^{2})}^{4}$

Step 4.22

Anything raised to $0$ is $1$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + 1 \cdot {(y^{2})}^{4}$

Step 4.23

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

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + {(y^{2})}^{4}$

Step 4.24

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

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Step 4.24.1

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

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + y^{2 \cdot 4}$

Step 4.24.2

Multiply $2$ by $4$ .

$16 x^{4} + 32 x^{3} y^{2} + 24 x^{2} y^{4} + 8 x y^{6} + y^{8}$