Trigonometry Examples

${(2 x + 5)}^{5}$

Step 1

Use the Binomial Theorem.

${(2 x)}^{5} + 5 {(2 x)}^{4} \cdot 5 + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $2 x$ .

$2^{5} x^{5} + 5 {(2 x)}^{4} \cdot 5 + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.2

Raise $2$ to the power of $5$ .

$32 x^{5} + 5 {(2 x)}^{4} \cdot 5 + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.3

Apply the product rule to $2 x$ .

$32 x^{5} + 5 (2^{4} x^{4}) \cdot 5 + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.4

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

$32 x^{5} + 5 (16 x^{4}) \cdot 5 + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.5

Multiply $16$ by $5$ .

$32 x^{5} + 80 x^{4} \cdot 5 + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.6

Multiply $5$ by $80$ .

$32 x^{5} + 400 x^{4} + 10 {(2 x)}^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.7

Apply the product rule to $2 x$ .

$32 x^{5} + 400 x^{4} + 10 (2^{3} x^{3}) \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.8

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

$32 x^{5} + 400 x^{4} + 10 (8 x^{3}) \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.9

Multiply $8$ by $10$ .

$32 x^{5} + 400 x^{4} + 80 x^{3} \cdot 5^{2} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.10

Raise $5$ to the power of $2$ .

$32 x^{5} + 400 x^{4} + 80 x^{3} \cdot 25 + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.11

Multiply $25$ by $80$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 10 {(2 x)}^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.12

Apply the product rule to $2 x$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 10 (2^{2} x^{2}) \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.13

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

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 10 (4 x^{2}) \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.14

Multiply $4$ by $10$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 40 x^{2} \cdot 5^{3} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.15

Raise $5$ to the power of $3$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 40 x^{2} \cdot 125 + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.16

Multiply $125$ by $40$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 5 (2 x) \cdot 5^{4} + 5^{5}$

Step 2.17

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

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Step 2.17.1

Move $5^{4}$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 5^{4} \cdot 5 (2 x) + 5^{5}$

Step 2.17.2

Multiply $5^{4}$ by $5$ .

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Step 2.17.2.1

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

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 5^{4} \cdot 5^{1} (2 x) + 5^{5}$

Step 2.17.2.2

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

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 5^{4 + 1} (2 x) + 5^{5}$

Step 2.17.3

Add $4$ and $1$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 5^{5} (2 x) + 5^{5}$

Step 2.18

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

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 3125 (2 x) + 5^{5}$

Step 2.19

Multiply $2$ by $3125$ .

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 6250 x + 5^{5}$

Step 2.20

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

$32 x^{5} + 400 x^{4} + 2000 x^{3} + 5000 x^{2} + 6250 x + 3125$