Precalculus Examples

$- 3 {(5 b + 3)}^{3}$

Step 1

Use the Binomial Theorem.

$- 3 ({(5 b)}^{3} + 3 {(5 b)}^{2} \cdot 3 + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $5 b$ .

$- 3 (5^{3} b^{3} + 3 {(5 b)}^{2} \cdot 3 + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2.1.2

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

$- 3 (125 b^{3} + 3 {(5 b)}^{2} \cdot 3 + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2.1.3

Apply the product rule to $5 b$ .

$- 3 (125 b^{3} + 3 (5^{2} b^{2}) \cdot 3 + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2.1.4

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

$- 3 (125 b^{3} + 3 (25 b^{2}) \cdot 3 + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2.1.5

Multiply $25$ by $3$ .

$- 3 (125 b^{3} + 75 b^{2} \cdot 3 + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2.1.6

Multiply $3$ by $75$ .

$- 3 (125 b^{3} + 225 b^{2} + 3 (5 b) \cdot 3^{2} + 3^{3})$

Step 2.1.7

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

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Step 2.1.7.1

Move $3^{2}$ .

$- 3 (125 b^{3} + 225 b^{2} + 3^{2} \cdot 3 (5 b) + 3^{3})$

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

$- 3 (125 b^{3} + 225 b^{2} + 3^{2} \cdot 3^{1} (5 b) + 3^{3})$

Step 2.1.7.2.2

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

$- 3 (125 b^{3} + 225 b^{2} + 3^{2 + 1} (5 b) + 3^{3})$

Step 2.1.7.3

Add $2$ and $1$ .

$- 3 (125 b^{3} + 225 b^{2} + 3^{3} (5 b) + 3^{3})$

Step 2.1.8

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

$- 3 (125 b^{3} + 225 b^{2} + 27 (5 b) + 3^{3})$

Step 2.1.9

Multiply $5$ by $27$ .

$- 3 (125 b^{3} + 225 b^{2} + 135 b + 3^{3})$

Step 2.1.10

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

$- 3 (125 b^{3} + 225 b^{2} + 135 b + 27)$

Step 2.2

Apply the distributive property.

$- 3 (125 b^{3}) - 3 (225 b^{2}) - 3 (135 b) - 3 \cdot 27$

Step 3

Simplify.

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Step 3.1

Multiply $125$ by $- 3$ .

$- 375 b^{3} - 3 (225 b^{2}) - 3 (135 b) - 3 \cdot 27$

Step 3.2

Multiply $225$ by $- 3$ .

$- 375 b^{3} - 675 b^{2} - 3 (135 b) - 3 \cdot 27$

Step 3.3

Multiply $135$ by $- 3$ .

$- 375 b^{3} - 675 b^{2} - 405 b - 3 \cdot 27$

Step 3.4

Multiply $- 3$ by $27$ .

$- 375 b^{3} - 675 b^{2} - 405 b - 81$