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Chapter 4: Problem 55

Simplify. $$ 14 \times(82.6+67.9) $$

Short Answer

Expert verified
The expression simplifies to 2107.

Step by step solution

01

Simplify Inside the Parentheses

First simplify the expression inside the parentheses. Add 82.6 and 67.9 together.\[ 82.6 + 67.9 = 150.5 \]
02

Multiply the Result

Next, multiply the result from Step 1 by 14.\[ 14 \times 150.5 \]
03

Perform the Multiplication

Multiply 14 by 150.5 to get the final result.\[ 14 \times 150.5 = 2107 \]

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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 a fundamental step in solving many mathematical problems. It means converting complex expressions into simpler, more manageable forms.
In this exercise, we simplify the expression inside the parentheses first, then perform additional operations.
The order in which we simplify expressions follows specific rules called the order of operations, which helps ensure consistency and accuracy.
Here, we see the expression:
For example, simplify:
\(14 \times(82.6+67.9)\).
First, we perform the addition within the parentheses.
addition
Addition is one of the basic arithmetic operations. It involves calculating the total of two or more numbers or quantities.
In our example, we add 82.6 and 67.9.
\[ 82.6 + 67.9 = 150.5 \]
Detailed steps:
  • Ensure the numbers are aligned by their decimal points.
  • Add each column from right to left.
  • Carry over any extra value to the next column.

Adding carefully will give us the sum needed for the next step of our calculation.
multiplication
Multiplication is another essential arithmetic operation. It involves calculating the product of two or more numbers.
In our example, after finding the sum of 82.6 and 67.9, which is 150.5, we multiply this result by 14.
We have:
\[ 14 \times 150.5 \]
When multiplying:
  • Break down the problem into more straightforward steps if necessary.
  • Multiply each digit of one number by each digit of the other and sum the partial results.
  • Keep track of where the decimal point will go in the final product.

Performing multiplication carefully will give us the accurate final result, which tells us the overall simplification of the initial expression.
order of operations
The order of operations is a set of rules that dictates the sequence in which calculations should be performed to ensure consistent results.
The common sequence is remembered by the acronym PEMDAS:
  • Parentheses
  • Exponents (i.e., powers and roots, etc.)
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)

Following this sequence is crucial, as it affects the outcome of the calculations.
In our example:
\[14 \times(82.6+67.9) \]
We start by simplifying inside the parentheses first, followed by multiplication, ensuring we get the correct answer of 2107.

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