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Chapter 1: Problem 80

Simplify the given expression. $$ 10+11\left[20-\left(2^{2}+1\right)\right] $$

Short Answer

Expert verified
The expression simplifies to 175.

Step by step solution

01

Simplify Innermost Parentheses

Begin by simplifying the expression inside the innermost parentheses: \( 2^2 + 1 = 4 + 1 = 5 \). So, the expression now becomes \( 10 + 11\left[20 - 5\right] \).
02

Simplify Expressions within Brackets

Next, simplify the expression inside the brackets:\( 20 - 5 = 15 \).This changes the expression to \( 10 + 11 \times 15 \).
03

Perform Multiplication

Proceed to perform the multiplication: \( 11 \times 15 = 165 \).This updates the expression to \( 10 + 165 \).
04

Add Remaining Terms

Finally, add the terms left in the expression: \( 10 + 165 = 175 \).

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Key Concepts

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

Innermost Parentheses
In an expression involving multiple parentheses or brackets, it's crucial to start simplifying from the innermost parentheses. This follows the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
To apply this, first look inside the deepest set of parentheses or brackets in your expression.
  • In our exercise, the innermost part is the expression within the parentheses: \(2^2 + 1\).
  • We tackle this section first by evaluating the power: \(2^2\), which equals 4.
  • Then, perform the addition inside: \(4 + 1\) = 5.
Once the innermost parentheses are simplified, you'll have a simpler expression to work with in the subsequent steps.
Simplification
Simplification is the process of reducing the expression to its simplest form by performing operations step-by-step.
In the exercise, after resolving the innermost parentheses, we observe the expression: \[10 + 11 [ 20 - 5 ]\]
This step involves solving what's inside the brackets:
  • Calculate \(20 - 5\) which simplifies to 15.
This simplification helps in making calculations easier as it reduces the complexity of the expression, setting the stage for subsequent operations like multiplication.
Multiplication
With the expression reduced, we now focus on multiplication, as it is next in the order of operations hierarchy after parentheses and exponents.
Looking at the simplified expression from before: \[10 + 11 \times 15\]
We proceed with the multiplication:
  • Multiply 11 by 15 to get 165.
Performing multiplication at the correct step is key, as skipping or rearranging this can lead to incorrect results. Now, the expression is further reduced, making the final process much more straightforward.
Addition
The final operation to perform is addition, which often tends to be the last step in the order of operations process. At this stage, our expression is: \[10 + 165\]
Simply add these numbers together:
  • \(10 + 165 = 175\)
When all other operations have been properly executed beforehand, addition serves to sum up the remaining numbers, delivering the final simplified and accurate result.

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