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Chapter 1: Problem 50

Do \(2 \times 4+3\) and \(2 \times(4+3)\) have the same value? Explain.

Short Answer

Expert verified
No, they do not have the same value; the first result is 11, and the second is 14.

Step by step solution

01

Solve the first expression

First, solve the expression \(2 \times 4 + 3\). According to the order of operations, multiplication is performed before addition. So, multiply 2 by 4, which gives 8, and then add 3. Therefore, the expression evaluates to \(8 + 3 = 11\).
02

Solve the second expression

Now, solve the expression \(2 \times (4 + 3)\). Start by solving the operation inside the parentheses: \(4 + 3 = 7\). Then multiply the result by 2: \(2 \times 7 = 14\).
03

Compare the results

Compare the results of the two expressions. The first expression evaluated to 11, and the second evaluated to 14. These results are not the same, indicating that \(2 \times 4 + 3\) and \(2 \times (4 + 3)\) do not have the same value.

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Key Concepts

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

Understanding Expressions
When we talk about expressions in mathematics, we are referring to combinations of numbers, variables, and operations. These operations might include addition, subtraction, multiplication, and division. In the example given, both \(2 \times 4 + 3\) and \(2 \times (4 + 3)\) are expressions.
  • Each expression is a group of terms that are combined to represent a particular value.
  • Understanding how to work with expressions is crucial, as it forms the basis of solving mathematical problems.

The main point to remember with expressions is that the way operations are prioritized can lead to different results, emphasizing the need for a clear understanding of the order of operations.
The Role of Parentheses
Parentheses play an essential role in mathematics by determining the order in which operations are performed in an expression. This is because operations within parentheses are always carried out first. In the expression \(2 \times (4 + 3)\), you must first calculate the value inside the parentheses, which is \(4 + 3 = 7\).
  • Parentheses help group specific parts of an expression that need to be addressed first.
  • They can significantly change the outcome of an expression, depending on where they are placed.

Understanding the function of parentheses is crucial to interpreting and solving expressions accurately.
Performing Multiplication
Multiplication in mathematics involves repeated addition. For instance, \(2 \times 4\) can be seen as adding 4 two times. It's one of the fundamental arithmetic operations and is typically solved before addition or subtraction unless parentheses dictate otherwise. In our example, after dealing with any parentheses, you proceed by multiplying:
  • First expression: \(2 \times 4 = 8\)
  • Second expression after simplifying in parentheses: \(2 \times 7 = 14\)

Correctly applying multiplication within expressions is key to reaching the correct result.
Understanding Addition
Addition is one of the basic operations used in expressions to combine values. It鈥檚 the final step of solving the first expression \(2 \times 4 + 3\), where after performing the multiplication, you add 3 to 8 resulting in 11. Addition is straightforward but crucial to get right in the context of order of operations.
  • Addition brings together the results of other operations, completing the value of the expression.
  • When combined with parentheses and multiplication, the order of how addition is performed can vary.

An accurate understanding of how to properly integrate addition into solving expressions is vital for success in mathematics.

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Most popular questions from this chapter

Simplify: \(15+(b+3)\)

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