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Chapter 1: Problem 91

Evaluate the given expression. Remember to follow the order of operations. $$3+2[3+2(3+2)]$$

Short Answer

Expert verified
29

Step by step solution

01

Evaluate Innermost Parentheses

Start by solving the expression within the innermost parentheses: \(3+2\). This simplifies to 5.
02

Multiply Within Brackets

Replace the innermost parentheses with 5 and evaluate the multiplication within the brackets: \(2(5)\). This simplifies to 10.
03

Evaluate Within Brackets

Now, replace \(2(3+2)\) with 10 in the original expression, resulting in: \(3+2[3+10]\). Evaluate the addition within the brackets next: \(3+10\). This simplifies to 13.
04

Final Multiplication

Now replace \(3+10\) with 13 in the expression: \(3+2[13]\). Multiply within the brackets: \(2 \times 13\). This simplifies to 26.
05

Final Addition

Finally, add 3 to the result from Step 4: \(3 + 26\). This simplifies to 29.

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Key Concepts

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

Parentheses
In mathematics, parentheses are used to group numbers and operations. This helps to clarify which operations should be performed first. Their main function is to override the normal order of operations.
In the expression given: 3 + 2[3 + 2(3 + 2)], we must first evaluate the innermost parentheses.
Following the order:
  • Simplify the expression inside the parentheses: 3+2 = 5
The use of parentheses directs us to perform 3+2 before any other operation. This grouping ensures that the calculation follows the intended sequence. Skipping this step or ignoring the grouping can lead to incorrect answers. Once the innermost parentheses are evaluated, continue outward, solving each grouped expression.
Multiplication
Multiplication is a fundamental arithmetic operation represented by an 'x' or sometimes a dot. It's essential to follow the order of operations where multiplication comes before addition.
In the expression: 3 + 2[3 + 2(3 + 2)], after simplifying the innermost parentheses to 5, we replace it in the expression:
  • 2(5) simplifies to 10
Multiplying the numbers within parentheses and brackets ensures that each step maintains the correct order of precedence. Continuing step-by-step like this helps in systematically breaking down complex expressions into simpler terms.
Addition
Addition is one of the basic operations in arithmetic, symbolized by '+'. According to the order of operations, addition is performed after parentheses and multiplication are evaluated.
In the final stages of our expression: 3 + 2[3 + 10], we have simplified it further by evaluating the brackets to get: 2 x 13 = 26.
Finally, adding the first term:
  • 3 + 26 simplifies to 29
Properly following the order of operations ensures that addition is done at the appropriate step, resulting in the correct answer.

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Most popular questions from this chapter

Describe each of the following expressions as a sum or a product. If the expression is a sum, list the terms. If the expression is a product, list the factors. (a) \(x+y+z\) (b) \(x y z\) (c) \(x y+z\) (d) \(x(y+z)\)

Evaluate the given expression. Remember to follow the order of operations. $$4+\\{3+5[2+2(3+1)]\\}$$

Compute the given expression. Round off your answer to two decimal places where necessary. $$28.7-32.56-18.61$$

Evaluate the given expression. Remember to follow the order of operations. $$30-(5 \cdot 4)-2$$

Fill in the appropriate ordering symbol: either \(<,>,\) or \(=\). 7_____7.
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