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Chapter 10: Problem 159

Multiply. $$-9 a(3 a+5)$$

Short Answer

Expert verified
-27a^2 - 45a

Step by step solution

01

- Distribute the Term

Distribute the \( -9a \) to each term inside the parentheses \( (3a + 5) \).
02

- Multiply the First Term

Multiply \( -9a \) by \( 3a \): \[ -9a \times 3a = -27a^2 \]
03

- Multiply the Second Term

Multiply \( -9a \) by \( 5 \): \[ -9a \times 5 = -45a \]
04

- Combine the Terms

Combine the results from Steps 2 and 3: \[ -27a^2 - 45a \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributing Terms
When you see an expression inside parentheses that needs to be multiplied by a term outside, you need to distribute this outside term to each term inside the parentheses individually.
For the exercise \(-9a (3a + 5)\), you start by distributing \(-9a\). Here鈥檚 how it works:
1. Distribute \(-9a\) to the first term inside the parentheses, which is \(3a\).
2. Then, distribute \(-9a\) to the second term inside the parentheses, which is 5.
This ensures that each term within the parentheses is multiplied by the term outside, and it's the key first step in solving these kinds of algebraic expressions.
Multiplying Algebraic Expressions
Next, we handle the multiplication process.
Let鈥檚 start with the first term: \(-9a\) multiplied by \(3a\). This can be broken down into:
\(-9 \times 3\) and \(a \times a\) which leads to:
\[-9a \times 3a = -27a^2\]
Now for the second term: \(-9a\) multiplied by 5. Here, the calculation is:
\[-9a \times 5 = -45a\]
It is essential to keep track of both the numerical and variable parts separately before combining them.
Combining Like Terms
After distributing and multiplying, the final step is to combine like terms. In this exercise, the resulting terms are \(-27a^2\) and \(-45a\).
Since there are no similar terms to combine further, the final expression remains:
\[-27a^2 - 45a\].
Combining like terms becomes crucial when you have multiple terms with the same variable and exponent. This simplifies the expression and makes it more concise.

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