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Chapter 10: Problem 4

Evaluate each expression. \(\frac{1}{2}(12)(7)\)

Short Answer

Expert verified
The expression evaluates to 42.

Step by step solution

01

Simplify the Expression

First, simplify the multiplication inside the parentheses. Here, you have to multiply 12 by 7. \[ 12 \times 7 = 84 \]
02

Apply the Fraction

Now, take the result from Step 1, which is 84, and multiply it by the fraction \( \frac{1}{2} \) to evaluate the expression.\[ \frac{1}{2} \times 84 = 42 \]

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Key Concepts

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

Simplifying Expressions
When working with algebraic expressions, simplifying means to make them easier to understand or solve. The key to simplifying expressions is to combine like terms and perform arithmetic operations as needed. In the process of simplification:
  • Identify and perform any arithmetic operations, such as multiplication or division, and organize the terms in the expression.
  • Combine like terms, which are terms that have identical variable parts and can be added or subtracted from each other.
  • Decide the operations to perform without altering the expression's value.
Simplifying expressions can be especially helpful when dealing with fractions and algebraic terms inside parentheses. When you see something like \( \frac{1}{2}(12)(7) \), your first step is to simplify the multiplication inside the parentheses: \(12 \times 7 = 84\). It makes the problem more manageable and prepares the expression for further operations.
Order of Operations
The importance of the order of operations in simplifying and evaluating expressions cannot be understated. It is a set way of performing mathematical calculations to arrive at the correct result. This sequence of operations is often remembered through the acronym PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
For instance, in the expression \( \frac{1}{2}(12)(7) \), the operation within the parentheses is completed first, according to PEMDAS. Thus, multiplying \(12\) by \(7\) is prioritized, simplifying the expression to \(84\). Following that, you follow through with division or multiplication by the fraction. Using PEMDAS ensures clarity and precision in mathematical problem-solving.
Evaluating Expressions
Evaluating an expression involves calculating its value once it is fully simplified. This process depends greatly on accurately following the steps of simplification and adhering to the correct order of operations. When evaluating:
  • Ensure all operations inside parentheses are complete.
  • Use the right sequence of arithmetic tasks.
  • Accurately perform mathematical operations specified in the expression.
In our example \( \frac{1}{2}(12)(7) \), once \(12\) and \(7\) are multiplied to give \(84\), the final step is to evaluate \( \frac{1}{2} \times 84 \), which results in \(42\). Evaluating expressions correctly yields a precise solution and is crucial in reaching the final and correct result of your calculations.

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