Question

lesson 23 | session 1 name
prepare for finding square roots and cube roots to solve problems
1 think about what you know about operations with rational numbers. fill in each box. use words, numbers, and pictures. show as many ideas as you can.
in my own words
my illustrations
product
examples
non - examples
which of the expressions at the right are examples of a product? explain.
2(10 + y)
x⁴
−7a
(3 + 4)(3 −

Explanation:

Step1: Define "product"

A product is the result of multiplying two or more quantities (numbers, variables, or expressions).

Step2: Analyze each expression

- For \(2(10 + y)\): This is the product of 2 and the expression \((10 + y)\) (using the distributive property idea, it's a multiplication of 2 with the sum inside the parentheses).
- For \(x^{4}\): This is \(x\times x\times x\times x\), which is a product of four \(x\)s. Wait, actually, in terms of the basic definition of a product as a multiplication of two or more factors, \(x^{4}\) is a power, but it can be seen as a product of repeated factors. However, let's check the other expressions too.
- For \(-7a\): This is the product of \(-7\) and \(a\) (multiplication of a number and a variable).
- For \((3 + 4)(3 - \dots)\) (assuming the last part is a typo, but the first part \((3 + 4)\) times another expression would be a product of two expressions: \((3 + 4)\) and the other factor). Wait, but let's re - evaluate.

Wait, actually, the key is to identify which expressions represent a multiplication of two or more quantities.

- \(2(10 + y)\): Multiplication of 2 and \((10 + y)\) → product.
- \(x^{4}\): This is a power, which is \(x\times x\times x\times x\), but sometimes in basic terms, when we talk about a product in the context of algebraic expressions, a product is an expression formed by multiplying two or more factors. However, \(x^{4}\) is a single - term power. Let's check \(-7a\): \(-7\times a\) → product. And \((3 + 4)(3 - \dots)\) (let's assume the full expression is \((3 + 4)(3 - b)\) for example) would be the product of \((3 + 4)\) and \((3 - b)\). But from the given expressions:

\(2(10 + y)\): It is \(2\times(10 + y)\), so it's a product of 2 and the binomial \((10 + y)\).

\(-7a\): It is \(-7\times a\), so it's a product of \(-7\) and \(a\).

\(x^{4}\): \(x^{4}=x\times x\times x\times x\), but in the context of simple algebraic expressions, when we talk about a product as an expression with a multiplication operation between two or more distinct factors (not just repeated factors in a power), \(x^{4}\) is a power, not a product in the same sense as the others. Wait, maybe the problem considers a product as an expression that is a result of multiplication of two or more factors (numbers, variables, or expressions). So:

- \(2(10 + y)\): Product (2 times \((10 + y)\))
- \(-7a\): Product (\(-7\) times \(a\))
- \((3 + 4)(3 - \dots)\): Product (the product of \((3 + 4)\) and the other factor)
- \(x^{4}\): Power, not a product in the basic sense of a multiplication of two or more different factors (it's a repeated multiplication of the same factor, but sometimes in basic algebra, a product is defined as an expression with a multiplication sign or implied multiplication between two or more factors).

But let's go back to the question: "Which of the expressions... are examples of a product? Explain."

So, let's analyze each:

1. \(2(10 + y)\): This is the product of 2 and \((10 + y)\) because it represents 2 multiplied by \((10 + y)\). The multiplication is either explicit (the 2 outside the parentheses implies multiplication) or can be seen as \(2\times(10 + y)\).

2. \(-7a\): This is the product of \(-7\) and \(a\) because it is equivalent to \(-7\times a\) (implied multiplication between the coefficient and the variable).

3. \(x^{4}\): This is a power, which is \(x\times x\times x\times x\), but in the context of this problem, when we talk about a product, we usually mean an expression that is the result of multiplying two or more different factors (not just repeated multiplication of the same factor). So \(x^{4}\) is not a product in the way the other expressions are.

4. \((3 + 4)(3 - \dots)\): Assuming the full expression is \((3 + 4)(3 - b)\) (for example), it is the product of \((3 + 4)\) and \((3 - b)\) because it represents the multiplication of these two binomial expressions.

Answer:

The expressions that are examples of a product are \(2(10 + y)\), \(-7a\), and \((3 + 4)(3 - \dots)\) (assuming the full expression is a product of two factors).

• For \(2(10 + y)\): It represents the multiplication of the number 2 and the algebraic expression \((10 + y)\), so it is a product.
• For \(-7a\): It represents the multiplication of the number \(-7\) and the variable \(a\), so it is a product.
• For \((3 + 4)(3 - \dots)\): It represents the multiplication of the expression \((3 + 4)\) (which simplifies to 7) and another algebraic expression (the part after the second parenthesis), so it is a product.

The expression \(x^{4}\) is a power (representing repeated multiplication of \(x\) by itself) and not a product in the sense of multiplying two or more distinct factors (a number and a variable, or two different algebraic expressions).