Algebra Examples

${(x + 3)}^{3}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Anything raised to $0$ is $1$ .

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

Step 4.3

Multiply $x^{3}$ by $1$ .

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

Step 4.4

Evaluate the exponent.

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

Step 4.5

Multiply $3$ by $3$ .

$x^{3} + 9 x^{2} + 3 \cdot {(x)}^{1} \cdot {(3)}^{2} + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.6

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 4.6.1

Move ${(3)}^{2}$ .

$x^{3} + 9 x^{2} + {(3)}^{2} \cdot 3 \cdot {(x)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.6.2

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

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Step 4.6.2.1

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

$x^{3} + 9 x^{2} + {(3)}^{2} \cdot 3^{1} \cdot {(x)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} + 9 x^{2} + 3^{2 + 1} \cdot {(x)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.6.3

Add $2$ and $1$ .

$x^{3} + 9 x^{2} + 3^{3} \cdot {(x)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.7

Simplify $3^{3} \cdot {(x)}^{1}$ .

$x^{3} + 9 x^{2} + 3^{3} \cdot x + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.8

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

$x^{3} + 9 x^{2} + 27 \cdot x + 1 \cdot {(x)}^{0} \cdot {(3)}^{3}$

Step 4.9

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

$x^{3} + 9 x^{2} + 27 x + {(x)}^{0} \cdot {(3)}^{3}$

Step 4.10

Anything raised to $0$ is $1$ .

$x^{3} + 9 x^{2} + 27 x + 1 \cdot {(3)}^{3}$

Step 4.11

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

$x^{3} + 9 x^{2} + 27 x + {(3)}^{3}$

Step 4.12

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

$x^{3} + 9 x^{2} + 27 x + 27$