Algebra Examples

{(m + 1)}^{3}

${(m + 1)}^{3}$

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})

${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(m)}^{3 - k} \cdot {(1)}^{k}

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(m)}^{3 - k} \cdot {(1)}^{k}$

Step 2

Expand the summation.

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

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

Step 3

Simplify the exponents for each term of the expansion.

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

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

Step 4

Simplify each term.

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Step 4.1

Multiply

1

$1$ by

{(1)}^{0}

${(1)}^{0}$ by adding the exponents.

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Step 4.1.1

Move

{(1)}^{0}

${(1)}^{0}$ .

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

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

Step 4.1.2

Multiply

{(1)}^{0}

${(1)}^{0}$ by

1

$1$ .

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Step 4.1.2.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 4.1.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.1.3

Add

0

$0$ and

1

$1$ .

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

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

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

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

Step 4.2

Simplify

1^{1} \cdot {(m)}^{3}

$1^{1} \cdot {(m)}^{3}$ .

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

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

Step 4.3

Evaluate the exponent.

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

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

Step 4.4

Multiply

3

$3$ by

1

$1$ .

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

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

Step 4.5

Simplify.

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

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

Step 4.6

One to any power is one.

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

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

Step 4.7

Multiply

3

$3$ by

1

$1$ .

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

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

Step 4.8

Multiply

1

$1$ by

{(1)}^{3}

${(1)}^{3}$ by adding the exponents.

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Step 4.8.1

Move

{(1)}^{3}

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

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

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

Step 4.8.2

Multiply

{(1)}^{3}

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

1

$1$ .

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Step 4.8.2.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 4.8.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.8.3

Add

3

$3$ and

1

$1$ .

m^{3} + 3 m^{2} + 3 m + 1^{4} \cdot {(m)}^{0}

$m^{3} + 3 m^{2} + 3 m + 1^{4} \cdot {(m)}^{0}$

m^{3} + 3 m^{2} + 3 m + 1^{4} \cdot {(m)}^{0}

$m^{3} + 3 m^{2} + 3 m + 1^{4} \cdot {(m)}^{0}$

Step 4.9

Simplify

1^{4} \cdot {(m)}^{0}

$1^{4} \cdot {(m)}^{0}$ .

m^{3} + 3 m^{2} + 3 m + 1^{4}

$m^{3} + 3 m^{2} + 3 m + 1^{4}$

Step 4.10

One to any power is one.

m^{3} + 3 m^{2} + 3 m + 1

$m^{3} + 3 m^{2} + 3 m + 1$

m^{3} + 3 m^{2} + 3 m + 1

$m^{3} + 3 m^{2} + 3 m + 1$