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Chapter 1: Problem 250

Simplify each expression. $$ 32-[5-(15-20)] $$

Short Answer

Expert verified
The simplified expression is 22.

Step by step solution

01

- Solve the innermost parentheses

Solve the expression inside the innermost parentheses first: \[15 - 20 = -5\]. So, the expression now becomes: \[32 - [5 - (-5)]\]
02

- Simplify the subtraction

Next, simplify the expression inside the brackets: \[5 - (-5)\] which is equal to \[5 + 5 = 10\]. Now, the expression becomes: \[32 - 10\]
03

- Final subtraction

Finally, subtract 10 from 32: \[32 - 10 = 22\]

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Key Concepts

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

Order of Operations
Understanding the order in which you should perform mathematical operations is vital for simplifying algebraic expressions correctly. This order is often remembered using the acronym PEMDAS, which stands for: Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right). This sequence ensures that everyone interprets and simplifies expressions in the same way.

For example, in the expression given in the exercise, following PEMDAS is crucial. We first simplify the innermost parentheses, then handle the remaining operations step-by-step. Always remember to:

  • Work from the innermost parentheses outward.
  • Perform any multiplications or divisions before additions or subtractions.
  • Proceed sequentially from left to right.
This way, the correct simplification and results are assured.
Parentheses
Parentheses play an essential role in grouping parts of an expression that need to be simplified first. They set apart specific terms or operations that should be handled before the rest. You might find nested parentheses (parentheses within parentheses), like in our example.

When dealing with nested parentheses:
  • Start simplifying from the innermost parentheses and move outward.
  • Replace the simplified part back into the expression to update it progressively.
In our example, the expression starts as \(32 - [5 - (15 - 20)]\). We first handle the innermost part, \(15 - 20\), which simplifies to \(-5\). Update the expression to \(32 - [5 - (-5)]\) and then address the next set of brackets. Parentheses help maintain the correct order of operations, ensuring your calculations are accurate.
Negative Numbers
Negative numbers can sometimes make simplification tricky, but they follow specific rules that make your work easier. Remember:
  • Subtracting a negative number is the same as adding the positive equivalent.
  • Adding a negative number is the same as subtracting the positive equivalent.
In our example, when we get to step 2, we face \(5 - (-5)\), which is equivalent to \(5 + 5\). This simplifies to 10. This rule converts complex looking but simple operations into straightforward additions or subtractions.

Understanding how to manage negative numbers is key to avoiding common mistakes. Always pay attention to the signs and apply these rules consistently to simplify expressions accurately.

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