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Chapter 5: Problem 10

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

Short Answer

Expert verified
-18r^3s^2

Step by step solution

01

Identify the terms

Identify the two terms that need to be multiplied: \(9r^3\) and \(-2s^2\).
02

Multiply the constants

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

Combine the variables

Since there are no common variables, simply write them together: \(r^3\) and \(s^2\).
04

Write the final result

Combine the result from step 2 and step 3: \(-18r^3s^2\)

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Key Concepts

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

Multiplying Monomials
Multiplying monomials involves a few simple steps. To start, it鈥檚 important to understand what a monomial is: a monomial is a single term that includes a coefficient (a number) and variables (letters). When you multiply monomials, you need to handle both the coefficients and the variables. Here鈥檚 a quick guide:
  • Identify each term.
  • Multiply the numerical coefficients.
  • Combine the variables.

In the provided exercise, you multiplied two monomials: \( 9r^3 \) and \( -2s^2 \). This step-by-step process ensures accurate results by addressing both parts of the monomial separately.
Variables
Variables are symbols that represent numbers; they are typically letters like \( r \) and \( s \). When multiplying monomials, if the variables don鈥檛 match, you simply write them next to each other. It鈥檚 important to note:
  • Variables with different bases (e.g., \( r \) and \( s \)) don't combine.
  • If variables are the same, you add their exponents.

In the exercise, since the variables are \( r \) and \( s \), they stay separate: \( r^3 \) and \( s^2 \). Understanding this concept helps in keeping the expression organized and correct.
Numerical Coefficients
Numerical coefficients are the numeric part of the terms and must be multiplied first. For example, in the exercise, you have the coefficients 9 and -2. You always:
  • Multiply the numbers as usual.
  • Keep the sign in mind (positive or negative).

For our example, we calculate \( 9 \times (-2) = -18 \). This step is crucial as it simplifies the expression by handling the numeric part separately before dealing with variables.

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Most popular questions from this chapter

Use scientific notation to calculate the answer to each problem. On February \(17,2009,\) Congress raised the U.S. government's debt limit to \(\$ 1.2 \times 10^{13} .\) When this national debt limit is reached, about how much will it be for every man, women, and child in the country? Use 300 million as the population of the United States.

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ \left(2 x+\frac{2}{3} y\right)\left(3 x-\frac{3}{4} y\right) $$

Multiply. $$ 4(2 a+6 b) $$

Assume that \(a\) is a number greater than 1 . Arrange the following terms in order from least to greatest: \(-(-a)^{3},-a^{3},(-a)^{4},-a^{4} .\) Explain how you decided on the order.

Perform each subtraction. $$8-(-4)$$
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