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Chapter 1: Problem 97

Use the distributive property to evaluate the given expression. $$ 6 \cdot(7-2) $$

Short Answer

Expert verified
The expression evaluates to 30.

Step by step solution

01

Identify the Expression

We are given the expression to evaluate: \(6 \cdot (7-2)\). Our task is to use the distributive property to simplify this.
02

Simplify Inside the Parentheses

First, simplify the expression inside the parentheses: \(7-2 = 5\). So now our expression looks like \(6 \cdot 5\).
03

Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). However, since there is only subtraction inside the parentheses, we deal directly with the multiplication: \(6 \cdot 5\).
04

Multiply to Get the Final Answer

Now, multiply the numbers to find the solution: \(6 \times 5 = 30\). So, the expression evaluates to 30.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is an essential part of solving math problems, especially when dealing with parentheses and operations like addition and subtraction.
It involves making the expression more straightforward while maintaining the original value.
  • Start by looking inside the parentheses. Operate on the numbers you find there before proceeding with any other calculations.
  • Remove any unnecessary elements like extra terms or similar factors to make the expression digestible.
In our example, we had the expression \(7-2\) inside the parentheses. Simplifying, we find that \(7-2\) equals 5. This step is important because it prepares us to tackle the rest of the expression more efficiently.
Order of Operations
The order of operations is a crucial guideline in mathematics that tells us how to carry out calculations systematically. It ensures that everyone evaluates expressions in a consistent manner.
Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • Parentheses: Always address operations inside parentheses first, as seen in the step where we simplified \(7-2\) to 5.
  • Exponents: Tackle any exponential terms next, although not present in this example.
  • Multiplication and Division: Proceed from left to right, performing these operations next. In our example, we multiply \(6\) by \(5\) after simplifying the parentheses.
  • Addition and Subtraction: Perform these last, again moving from left to right.
Applying this order helps us arrive at the right solution, ensuring we interpret every part of the mathematical expression correctly.
Multiplication
Multiplication is a fundamental operation in math and often appears in combination with other operations like addition or subtraction. It involves finding the total number of objects in equal-sized groups.
Here are some key points:
  • Think of multiplication as repeated addition. For example, \(6 \times 5\) is the same as adding \(6\) five times: \(6 + 6 + 6 + 6 + 6\).
  • Use multiplication to scale quantities, making it a versatile tool in math, science, and everyday calculations.
In our example, after simplifying to \(6 \cdot 5\), we multiplied directly for a result of 30. This approach highlights how vital multiplication is in understanding and evaluating expressions accurately.

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