Precalculus Examples

${(5 x + 5)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(5 x)}^{4 - k} \cdot {(5)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} \cdot {(5 x)}^{4 - 0} \cdot {(5)}^{0} + \frac{4!}{(4 - 1)! 1!} \cdot {(5 x)}^{4 - 1} \cdot {(5)}^{1} + \frac{4!}{(4 - 2)! 2!} \cdot {(5 x)}^{4 - 2} \cdot {(5)}^{2} + \frac{4!}{(4 - 3)! 3!} \cdot {(5 x)}^{4 - 3} \cdot {(5)}^{3} + \frac{4!}{(4 - 4)! 4!} \cdot {(5 x)}^{4 - 4} \cdot {(5)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(5 x)}^{4} \cdot {(5)}^{0} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(5 x)}^{4}$ by $1$ .

${(5 x)}^{4} \cdot {(5)}^{0} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.2

Apply the product rule to $5 x$ .

$5^{4} x^{4} \cdot {(5)}^{0} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.3

Multiply $5^{4}$ by ${(5)}^{0}$ by adding the exponents.

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Step 4.3.1

Move ${(5)}^{0}$ .

${(5)}^{0} \cdot 5^{4} x^{4} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.3.2

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

$5^{0 + 4} x^{4} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.3.3

Add $0$ and $4$ .

$5^{4} x^{4} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.4

Raise $5$ to the power of $4$ .

$625 x^{4} + 4 \cdot {(5 x)}^{3} \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.5

Apply the product rule to $5 x$ .

$625 x^{4} + 4 \cdot (5^{3} x^{3}) \cdot {(5)}^{1} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.6

Multiply $5^{3}$ by ${(5)}^{1}$ by adding the exponents.

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Step 4.6.1

Move ${(5)}^{1}$ .

$625 x^{4} + 4 \cdot ({(5)}^{1} \cdot 5^{3} x^{3}) + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.6.2

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

$625 x^{4} + 4 \cdot (5^{1 + 3} x^{3}) + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.6.3

Add $1$ and $3$ .

$625 x^{4} + 4 \cdot (5^{4} x^{3}) + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.7

Raise $5$ to the power of $4$ .

$625 x^{4} + 4 \cdot (625 x^{3}) + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.8

Multiply $625$ by $4$ .

$625 x^{4} + 2500 x^{3} + 6 \cdot {(5 x)}^{2} \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.9

Apply the product rule to $5 x$ .

$625 x^{4} + 2500 x^{3} + 6 \cdot (5^{2} x^{2}) \cdot {(5)}^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.10

Multiply $5^{2}$ by ${(5)}^{2}$ by adding the exponents.

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Step 4.10.1

Move ${(5)}^{2}$ .

$625 x^{4} + 2500 x^{3} + 6 \cdot ({(5)}^{2} \cdot 5^{2} x^{2}) + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.10.2

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

$625 x^{4} + 2500 x^{3} + 6 \cdot (5^{2 + 2} x^{2}) + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.10.3

Add $2$ and $2$ .

$625 x^{4} + 2500 x^{3} + 6 \cdot (5^{4} x^{2}) + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.11

Raise $5$ to the power of $4$ .

$625 x^{4} + 2500 x^{3} + 6 \cdot (625 x^{2}) + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.12

Multiply $625$ by $6$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot {(5 x)}^{1} \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.13

Simplify.

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot (5 x) \cdot {(5)}^{3} + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.14

Multiply $5$ by ${(5)}^{3}$ by adding the exponents.

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Step 4.14.1

Move ${(5)}^{3}$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot ({(5)}^{3} \cdot 5 x) + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.14.2

Multiply ${(5)}^{3}$ by $5$ .

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Step 4.14.2.1

Raise $5$ to the power of $1$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot ({(5)}^{3} \cdot 5^{1} x) + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.14.2.2

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

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot (5^{3 + 1} x) + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.14.3

Add $3$ and $1$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot (5^{4} x) + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.15

Raise $5$ to the power of $4$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 4 \cdot (625 x) + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.16

Multiply $625$ by $4$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + 1 \cdot {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.17

Multiply ${(5 x)}^{0}$ by $1$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + {(5 x)}^{0} \cdot {(5)}^{4}$

Step 4.18

Apply the product rule to $5 x$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + 5^{0} x^{0} \cdot {(5)}^{4}$

Step 4.19

Multiply $5^{0}$ by ${(5)}^{4}$ by adding the exponents.

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Step 4.19.1

Move ${(5)}^{4}$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + {(5)}^{4} \cdot 5^{0} x^{0}$

Step 4.19.2

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

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + 5^{4 + 0} x^{0}$

Step 4.19.3

Add $4$ and $0$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + 5^{4} x^{0}$

Step 4.20

Simplify $5^{4} x^{0}$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + 5^{4}$

Step 4.21

Raise $5$ to the power of $4$ .

$625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + 625$