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Chapter 4: Problem 10

Find each product. $$ -3 m^{6}\left(-5 m^{4}\right) $$

Short Answer

Expert verified
15m^{10}

Step by step solution

01

Multiply the coefficients

First, multiply the numerical coefficients together. Multiply (-3) by (-5): \[ (-3) \times (-5) = 15 \]
02

Multiply the variables

Next, multiply the variables. When multiplying variables with exponents, add the exponents together. Here we have \(m^6\) and \(m^4\): \[ m^6 \times m^4 = m^{6+4} = m^{10} \]
03

Combine the results

Now, combine the results from Step 1 and Step 2. The product is: \[ 15m^{10} \]

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Key Concepts

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

Coefficients
When multiplying polynomials, we start by multiplying the coefficients. Coefficients are the numerical parts of the terms. For instance, in the term \(-3m^6\), the coefficient is \-3\. Similarly, in \(-5m^4\), the coefficient is \-5\.
To multiply these coefficients, you simply multiply the numbers together: \(-3 \times -5 = 15\).
Notice that the multiplication of two negative numbers results in a positive number. This is an essential rule of arithmetic that should be remembered throughout your studies.
Exponents
After dealing with the coefficients, the next step is to focus on the variables with their exponents. Exponents represent how many times a number (the base) is multiplied by itself. For example, in the expression \(m^6\), the variable \m\ is raised to the exponent 6, meaning \(m \times m \times m \times m \times m \times m\).
When multiplying variables with exponents, the exponents are added together. This rule comes from the properties of exponents: \a^m \times a^n = a^{m+n}\.
In our exercise, we have \(m^6 \) and \(m^4\), so when multiplied, they become: \(m^6 \times m^4 = m^{6+4} = m^{10}\).
Variables
Variables are symbols used to represent numbers in algebraic expressions. They are usually letters like \(x, y, m\), etc. In our example, the variable is \m\.
Understanding how to work with variables is crucial. When you multiply polynomials, each term's variable must be correctly manipulated.
By recognizing and applying the rules for coefficients and exponents, you can simplify expressions involving variables. This knowledge is foundational for solving more complex algebraic problems.

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Most popular questions from this chapter

If one "dog" year is estimated to be about seven "human" years, the monomial \(7 x\) gives the dog's age in human years for \(x\) dog years. Evaluate this monomial for \(x=9\). Use the result to fill in the blanks: If a dog is ______ in dog years, then it is _________ in human years.

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-9\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$ $$

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Perform each indicated operation. \(\left(4 m^{2}-3 m+2\right)+\left(5 m^{2}+13 m-4\right)+\left(-16 m^{2}-4 m+3\right)\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5 k^{2}\left(k^{3}-3\right)\left(k^{2}-k+4\right) $$
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