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Chapter 1: Problem 781

Simplify using the distributive property. $$ 8(4 y+9) $$

Short Answer

Expert verified
The simplified expression is \(32y + 72\).

Step by step solution

01

Identify the Distributive Property

The distributive property is given by the formula: \(a(b + c) = ab + ac\). In this problem, we have \(8(4y + 9)\).
02

Apply the Distributive Property

Multiply each term inside the parentheses by the number outside the parentheses: \(8 \times 4y\) and \(8 \times 9\).
03

Perform the Multiplications

Calculate each product: \(8 \times 4y = 32y\) and \(8 \times 9 = 72\).
04

Combine the Results

Combine the results from the previous steps: \(32y + 72\).

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Key Concepts

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

simplifying expressions
When we talk about simplifying expressions, what we mean is to make the expression as straightforward as possible. This involves combining like terms and performing mathematical operations to reduce an expression to its simplest form.
For example, in the exercise given, simplifying means using the distributive property to break down and then add or multiply terms until we can't simplify any further. In other words, we took the term inside the parentheses and carefully expanded it by multiplying, and then combined our results. The simplified form of the expression $$ 8(4y + 9) = 32y + 72 $$.
algebraic multiplication
Algebraic multiplication involves distributing multiplication across addition or subtraction operations within an expression. When you see something like \[ 8(4y + 9) \] you use the distributive property to multiply each term inside the parentheses by the number outside.
Here's how it works step-by-step:
  • First, we recognize that we need to multiply 8 by both 4y and 9.
  • Next, we do the multiplication:
    \[ 8 \times 4y = 32y \] and \[ 8 \times 9 = 72 \].
  • Lastly, we combine them back together to get our final simplified expression:
    • \[ 32y + 72 \].
This step-by-step method ensures that each term inside the parentheses is correctly multiplied by the outside number, making your expression fully simplified.
basic algebra
Basic algebra is the foundation of higher mathematics and involves using symbols (like variables) to represent numbers. This allows us to write general expressions and equations that can be solved. Learning basic algebra starts with understanding simple concepts like variables, constants, and arithmetic operations.
In the original exercise, basic algebra concepts were used to simplify the expression. We identified the terms and used the distributive property to perform the calculations.
Key steps in any basic algebra problem include:
  • Identifying variables and constants: Here, y is a variable, and 8 and 9 are constants.
  • Applying arithmetic operations: These are addition, subtraction, multiplication, and division.
  • Using properties like the distributive property: This helps to break down more complex expressions into simpler parts.
By mastering these basic concepts, you're well on your way to understanding more advanced topics in algebra.

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