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Chapter 2: Problem 66

Review operations with real numbers. Simplify. $$ 3-2+5 \cdot 10 \div 5^{2} \cdot 2 $$

Short Answer

Expert verified
5

Step by step solution

01

Simplify exponents

Identify and simplify any exponents in the expression first. \( 5^{2} = 25 \). So the expression becomes: 3-2+5 \cdot 10 \div 25 \cdot 2
02

Perform multiplication and division from left to right

Start from the left and perform the multiplication and division: 1. \( 5 \cdot 10 = 50 \) 2. \( 50 \div 25 = 2 \) 3. \( 2 \cdot 2 = 4 \). Now the expression simplifies to: 3-2+4
03

Perform addition and subtraction from left to right

Finally, perform the addition and subtraction from left to right: 1. \( 3 - 2 = 1 \) 2. \( 1 + 4 = 5 \). So the entire expression simplifies to 5.

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Key Concepts

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

Simplifying Exponents
When simplifying expressions with exponents, it's crucial to handle the exponents first. This is part of the rule called the 'order of operations.' Exponents tell us how many times to multiply a number by itself. For example, in the expression \(5^2\), the exponent 2 means we multiply 5 by itself: \(5 \times 5 = 25\).
So, in the original exercise, we simplified \(5^2\) to get 25.
Order of Operations
The order of operations is a set of rules that dictate the sequence in which we perform mathematical operations. You might have heard of PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
By following these rules, we ensure that everyone simplifies expressions the same way. In the exercise, after simplifying the exponent, we performed multiplication and division before addressing addition and subtraction.
Multiplication and Division
Multiplication and division should be done from left to right, after handling any exponents. This step is crucial in maintaining the correct order and getting an accurate result. In the exercise, once we had \(3 - 2 + 5 \times 10 \div 25 \times 2\), we performed the operations as follows:
  • \(5 \times 10 = 50\)
  • \(50 \div 25 = 2\)
  • \(2 \times 2 = 4\)

This simplified the expression to \(3 - 2 + 4\).
Addition and Subtraction
Lastly, we perform addition and subtraction from left to right, just like with multiplication and division. It鈥檚 important to do these operations in order without skipping steps, to ensure the final answer is correct. In the exercise, we had \(3 - 2 + 4\). Following the order, we proceeded with the calculations as:
  • \(3 - 2 = 1\)
  • \(1 + 4 = 5\)

Thus, the entire expression simplified to 5.

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