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Chapter 10: Problem 19

Multiply. $$(3 y)(4 y)$$

Short Answer

Expert verified
12y^2

Step by step solution

01

- Identify the terms

Look at the two terms that need to be multiplied: 1. \(3y\)2. \(4y\)
02

- Multiply the coefficients

Multiply the numerical coefficients of the terms: \(3\) and \(4\).This gives: \[3 \times 4 = 12\]
03

- Multiply the variables

Next, multiply the variable parts. Since both terms have the same variable \(y\), use the property of exponents:\(y \times y = y^2\)
04

- Combine the results

Combine the result of the coefficients and the variables:\(12 \times y^2 = 12y^2\)

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Key Concepts

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

Coefficients
When multiplying algebraic terms, it's important to understand the role of coefficients. Coefficients are the numerical parts of terms. In the given problem, the coefficients are 3 and 4 from the terms \(3y\) and \(4y\) respectively.
Multiplying coefficients is straightforward: simply multiply the two numbers:
\[3 \times 4 = 12\]
This step simplifies our problem and sets the foundation for combining with variables and exponents.
Variables
Variables represent unknown quantities and are usually denoted by letters like \(x\), \(y\), or \(z\). In our problem, the variable is \(y\).
When multiplying terms with the same variable, we combine these variables. For instance, in the problem, \(3y\) and \(4y\) both have \(y\).
Combining these gives:
  • \(y \times y = y^2\)

This shows that two variables of the same kind multiply together to form a new variable expression with an exponent.
Exponents
Exponents tell us how many times a number or a variable is multiplied by itself. In our context, \(y^2\) means that \(y\) is multiplied by itself.
When multiplying variables with exponents, if the base (the variable part) is the same, we add the exponents.
  • For \(y^1 \times y^1 = y^{1+1} = y^2\)

Here, we started with \(y\) and \(y\), both with an implicit exponent of 1. Multiplying them led to adding their exponents: 1 + 1. Therefore, \(y \times y = y^2\). Combining coefficients and variables with added exponents gives us the final term: \(12y^2\).

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Most popular questions from this chapter

Perform the indicated operations. $$\left(\frac{7}{8} a-4 b-\frac{1}{2} c\right)-\left(-\frac{1}{4} a+\frac{1}{2} b+\frac{3}{8} c\right)$$

Simplify. Write the answers with positive exponents only. $$z^{-4} z^{7}$$

Simplify the expression. $$(2 w)^{5}$$

Simplify. Write the answers with positive exponents only. $$\left(p^{4}\right)^{-5}$$

Simplify. Write the answer with positive exponents only. $$\frac{5}{a^{-2} b^{-4}}$$
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