Simplify expressions by combining like terms and then use the distributive property.5b9(4ab5 + 3ab5 − b3)

1. Break Down the Problem

We need to simplify the expression $5b^9(4ab^5 + 3ab^5 - b^3)$ by first combining like terms inside the parentheses and then using the distributive property.

2. Relevant Concepts

• Combining Like Terms: In algebra, like terms are terms that have the same variables raised to the same powers.
• Distributive Property: This property states that $a(b + c) = ab + ac$.

3. Analysis and Detail

Step 1: Combine Like Terms

First, combine the terms in the parentheses:

$4ab^5 + 3ab^5 = (4 + 3)ab^5 = 7ab^5$

So, the expression now becomes:

$5b^9(7ab^5 - b^3)$

Step 2: Apply the Distributive Property

Now, use the distributive property to multiply $5b^9$ with each term inside the parentheses:

$5b^9 \cdot 7ab^5 - 5b^9 \cdot b^3$

Calculating each term:

1. $5b^9 \cdot 7ab^5 = 35ab^{14}$ (since $b^9 \cdot b^5 = b^{14}$)
2. $5b^9 \cdot b^3 = 5b^{12}$ (since $b^9 \cdot b^3 = b^{12}$)

Putting it all together:

$35ab^{14} - 5b^{12}$

4. Verify and Summarize

Verify the calculations:

1. Combining $4ab^5$ and $3ab^5$ gives $7ab^5$.
2. The multiplication using the distributive property has been applied correctly.

Final Answer

The simplified expression is:

$35ab^{14} - 5b^{12}$