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Magazine Chapter 2: Problem 64
For Exercises \(1-74,\) simplify. $$-3\\{14-2|20-7(4)|\\}+[(3)(-9)-(-21)]^{2}$$
Short Answer
Expert verified
Question: Simplify the given expression: $$-3\\{14-2|20-7(4)|\\}+[(3)(-9)-(-21)]^{2}$$
Answer: 162
Step by step solution
01
Evaluate the innermost parentheses
First, let's look at the absolute value within the braces: \(|20-7(4)|\). We multiply 7 with 4, and subtract the result from 20.
$$|-28| = 28$$
Now, the expression becomes:
$$-3\\{14-2(28)\\}+[(3)(-9)-(-21)]^{2}$$
02
Evaluate the braces
Next, let's perform the remaining operation within the braces: \(14-2(28)\). We multiply 2 with 28, and subtract the result from 14.
$$14-56 = -42$$
Now, the expression becomes:
$$-3(-42)+[(3)(-9)-(-21)]^{2}$$
03
Evaluate the remaining parentheses
Now, let's perform the operations within the remaining parentheses: \((3)(-9)-(-21)\). First, we multiply 3 with -9, then add 21 to the result.
$$-27+21 = -6$$
Now, the expression becomes:
$$-3(-42)+(-6)^{2}$$
04
Evaluate the exponent
Next, let's evaluate the exponent: \((-6)^{2}\). We raise -6 to the power of 2.
$$(-6)^{2} = 36$$
Now, the expression becomes:
$$-3(-42)+36$$
05
Perform multiplication
Now, let's perform the multiplication: \(-3(-42)\). We multiply -3 with -42.
$$-3(-42) = 126$$
Now, the expression becomes:
$$126+36$$
06
Perform addition
Finally, let's perform the addition operation: \(126+36\). We add 126 and 36.
$$126+36 = 162$$
The simplified expression is:
$$162$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value
Absolute value refers to the distance of a number from zero on a number line. It is always non-negative, as it represents magnitude without direction. In this exercise, we encountered an absolute value within the expression \( |20-7(4)| \). The steps are as follows:
- Calculate inside the absolute value: substitute and simplify \( 20-7(4) \) to get \( -28 \).
- The absolute value of \( -28 \) is \( 28 \), since absolute value disregards the negative sign.
Order of Operations
The order of operations is a fundamental rule dictating the sequence in which parts of a mathematical expression are solved. Known by the acronym PEMDAS:
- Parentheses
- Exponents
- Multiplication and Division
- Addition and Subtraction
Exponents
Exponents represent repeated multiplication of a number by itself. In this exercise, we had to evaluate \( (-6)^2 \). Here's a breakdown of the process:
- The expression \( (-6)^2 \) means \( -6 \) multiplied by itself: \( (-6) \times (-6) \).
- This results in \( 36 \), recognizing that the product of two negative numbers is positive.
Multiplication
Multiplication involves combining repeated additions. We applied multiplication several times in this exercise:
- Calculating \( 7 \times 4 \) and \( 2 \times 28 \) within the absolute value and braces respectively.
- Handling \( -3 \times (-42) \), translating two negatives into a positive result of \( 126 \).
Addition
Addition is the basic arithmetic operation of combining numbers to find a total. In the final steps of our task, we performed:
- Addition of the two terms \( 126 + 36 \), leading to the final result of \( 162 \).
