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Chapter 1: Problem 60

Find each product or quotient. $$ 8\left(\frac{y}{8}\right) $$

Short Answer

Expert verified
y

Step by step solution

01

Identify the Expression

The given expression is \( 8\frac{y}{8} \). It involves both multiplication and division.
02

Simplify the Expression

Notice that multiplication and division are inverse operations. In the expression \( 8 \times \frac{y}{8} \), the 8s will cancel out each other.
03

Perform the Operations

Since \(8 \times \frac{1}{8} = 1\), the expression simplifies to \(y\).
04

Final Answer

Thus, after simplification, the expression \( 8\frac{y}{8} \) results in \(y\).

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Key Concepts

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

multiplication and division
Let's talk about the concepts of multiplication and division in algebraic expressions. When you see a term like \(8 \left(\frac{y}{8}\right)\), it's important to realize that multiplication and division are being used together. Multiplication and division are inverse operations. This means they can undo each other. When you multiply a number by 8 and then divide it by 8, you get the original number back.

Here鈥檚 a simpler example - consider the expression \(3 \left(\frac{x}{3}\right)\):
  • Multiply the variable x by 3.
  • Then, divide the result by 3.
  • The multiplication and division cancel out, leaving you with x.
This is why multiplication and division are powerful tools in algebra for simplifying expressions.
inverse operations
Inverse operations are operations that reverse the effect of each other. In our original problem, the expression is \(8 \left(\frac{y}{8}\right)\). Here, multiplication by 8 and division by 8 are inverse operations. They cancel each other out.

Think of inverse operations like a see-saw:
  • If you move up on one side (multiply), moving down an equal amount on the other side (divide) brings you back to the start.
  • Examples include addition/subtraction and multiplication/division.
Understanding that these operations can cancel each other helps in simplifying complex expressions more intuitively.
simplification
Simplification is all about making expressions easier to work with. When you simplify an expression, you perform operations step by step until you get the simplest form. For \(8 \left(\frac{y}{8}\right)\):
  • First, recognize that multiplication and division are present.
  • Next, apply inverse operations.
  • Multiplication by 8 and division by 8 cancel each other, simplifying the expression to y.
Thus, the simplified form of the expression is just y. Simplification makes it easier to understand and solve algebraic problems.

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