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Chapter 4: Problem 11

Find each product. $$ 5 p\left(3 q^{2}\right) $$

Short Answer

Expert verified
15pq^{2}

Step by step solution

01

Identify the terms

Recognize the terms that need to be multiplied together: 5p and \(3q^{2}\).
02

Apply the multiplication

Multiply the constants and the variables separately. Here, multiply the constants 5 and 3 to get 15. Then, combine the variables by keeping p and \(q^{2}\) together.
03

Combine the results

Combine the results of the multiplication. The product of the constants is 15, and the variables stay as they are. So, the final product is \(15pq^{2}\).

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Key Concepts

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

variable multiplication
When we talk about variable multiplication in algebra, we mean multiplying one or more variables together. Variables are symbols like p, q, x, or y.
They represent unknown numbers. To keep it simple, let's consider the example of the exercise given: multiplying 5p and \(3q^2\).
Here's what happens step-by-step:
  • First, you identify the variables being used, which are p and \(q^2\).
  • Second, you multiply any numerical constants (like 5 and 3 in this case).
  • Third, you combine these constants with the variables to get the final expression \(15pq^2\).
It's important to remember that the variables should be kept as part of the final product unless they have the same base and can be combined.
This concept is foundational, as you will use variable multiplication in more complex algebraic expressions.
constants
In algebra, constants are the numbers without any variables. They are fixed values that do not change. For example, in the exercise, the constants are 5 and 3.
These numbers remain the same in any calculation. When multiplying constants:
  • Simply perform the usual arithmetic multiplication.
In the given exercise, multiply 5 and 3 to get 15.
Unlike variables, constants don鈥檛 alter during multiplication. If you have an expression that involves both constants and variables, always handle the constants first.
After multiplying the constants, incorporate the variables into the final product.
For instance, in \(5p(3q^2)\), multiply the constants first, then attach the resulting constant to the variables to conclude with \(15pq^2\).
algebraic expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation signs. Expressions like \(5p(3q^2)\) are examples of this.
Let's break it down:
  • The term 5p is an algebraic term with a constant (5) and a variable (p).
  • Similarly, \(3q^2\) consists of a constant (3) and a variable raised to a power (q squared).
When you multiply such expressions, each part鈥攂oth constants and variables鈥攏eeds special attention.
Identify each type of term, follow mathematical operations correctly, and put them together in the end.
In our example, start by identifying the constant terms and variables, then apply the multiplication rules: multiply constants first, then combine with variables.
The result \(15pq^2\) merges all parts of the original algebraic expression correctly.

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Most popular questions from this chapter

Add or subtract as indicated. \(\left(3 k^{2} h^{3}+5 k h+6 k^{3} h^{2}\right)-\left(2 k^{2} h^{3}-9 k h+k^{3} h^{2}\right)\)

Perform each indicated operation. \(\left(16 x^{3}-x^{2}+3 x\right)+\left(-12 x^{3}+3 x^{2}+2 x\right)\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ -4 x^{3}\left(3 x^{4}+2 x^{2}-x\right)(-2 x+1) $$

\begin{aligned} &\text { Find the difference of the sum of } 5 x^{2}+2 x-3 \text { and } x^{2}-8 x+2 \text { and the sum of } 7 x^{2}-3 x+6\\\ &\text { and }-x^{2}+4 x-6 \end{aligned}

Find each product. $$ (4 x+3)(2 y-1) $$
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