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Chapter 4: Problem 14

Find each product. $$ 9 r^{3}\left(-2 s^{2}\right) $$

Short Answer

Expert verified
-18r^{3}s^{2}

Step by step solution

01

- Identify the Terms

The given product is composed of two terms: \(9r^{3}\) and \(-2s^{2}\).
02

- Multiply the Coefficients

Multiply the coefficients of the terms: \(9\) and \(-2\). \[ 9 \times (-2) = -18 \]
03

- Combine the Variables

Since the terms \(r^{3}\) and \(s^{2}\) have different bases, simply write them together: \[ r^{3} \times s^{2} = r^{3}s^{2} \]
04

- Write the Final Product

Combine the result from coefficients and the variables to write the final expression: \[ -18r^{3}s^{2} \]

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Key Concepts

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

Coefficients
Let's delve into the concept of coefficients in algebra. Coefficients are the numerical part of terms in an algebraic expression. They are the numbers that multiply the variables. In our exercise, the coefficients are 9 and -2. When multiplying algebraic expressions, you first multiply the coefficients together. In this case, multiplying 9 and -2 gives us -18. Remember: * Coefficients can be positive or negative. * Always multiply coefficients first before dealing with variables.
Variables
Now, let's look at variables. Variables are symbols, usually letters, that represent unknown numbers. They are essential in algebra and can have different bases. For example, in our exercise, the variables are \( r^{3} \) and \( s^{2} \). These variables have different bases ('r' and 's'), so they are combined together in the final expression as \( r^{3} s^{2} \). Key Points to Remember: * Variables can be combined only if they have the same base. * If they have different bases, simply write them together.
Exponents
Lastly, let's explore exponents. Exponents indicate how many times a variable is multiplied by itself. In our exercise, the exponents are 3 and 2 in \( r^{3} \) and \( s^{2} \) respectively. When multiplying expressions with the same base, you add the exponents. However, since our variables have different bases, we just write their exponents as they are. Important Tips: * Exponents follow specific rules when multiplying and dividing terms. * For terms with the same base: \( x^{a} \times x^{b} = x^{a+b} \). * For terms with different bases, just keep the exponents with their respective variables.

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Most popular questions from this chapter

During the \(2016-2017\) season, Broadway shows grossed a total of \(\$ 1.45 \times 10^{9}\). Total attendance for the season was \(1.33 \times 10^{7}\). What was the average ticket price (to the nearest cent) for a Broadway show? (Data from The Broadway League.)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (5 k+3 q)^{2} $$

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Perform each indicated operation. \(\left(x^{2}+x\right)-\left(3 x^{2}+2 x-1\right)\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m-5)^{3} $$
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