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Chapter 2: Problem 31

For Exercises \(1-74,\) simplify. $$9 \div 3-(-2)^{2}$$

Short Answer

Expert verified
$$ Answer: The simplified form of the expression is $$-1$$.

Step by step solution

01

Solve the exponent

We first simplify the exponent part of the expression. Given the expression, we can see that the exponent is \((-2)^{2}\). $$(-2)^{2} = (-2) \cdot (-2) = 4$$ Now the expression becomes: $$9 \div 3 - 4$$
02

Perform the division

Next, we perform the division operation in the expression. Given the expression, we can see that the division involves \(9 \div 3\). $$9 \div 3 = 3$$ Now the expression becomes: $$3 - 4$$
03

Solve the subtraction

Finally, we solve the subtraction part of the expression, which is now \(3 - 4\): $$3 - 4 = -1$$ The simplified expression is: $$-1$$

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Key Concepts

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

Order of Operations
The order of operations is a vital rule in mathematics that dictates the correct sequence in which to solve parts of a mathematical expression. This ensures everyone arrives at the same answer. The standard order we follow is often remembered by the acronym PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our exercise, we applied this rule by first dealing with the exponent, then the division, and finally the subtraction. Remembering the PEMDAS rule helps in solving mathematical problems without errors.
Exponents
Exponents are a shorthand for repeated multiplication of a number by itself. For example, \((-2)^{2}\) means multiplying -2 by itself: \((-2) imes (-2) = 4\).
Exponents make calculations easier and expressions neater. They often appear in algebra, and understanding them is essential for advanced mathematics topics. Since \((-2) \times (-2)\) involves multiplication of two negative numbers, the result is positive, making \(4\) the result of \((-2)^{2}\).
In our exercise, calculating the exponent correctly was important as a first step before tackling division and subtraction.
Division
Division is the process of splitting a number into equal parts or groups. It is one of the fundamental arithmetic operations and is shown using the symbol \(\div\) or a slash \(/\).
In the exercise, we calculated \(9 \div 3\), which involves dividing 9 into 3 equal parts. The result is \(3\), since 3 times 3 equals 9.
It's important to perform division before moving on to addition or subtraction, as prescribed by the order of operations. In our problem, solving \(9 \div 3\) before subtracting ensures the accuracy of the solution.
Subtraction
Subtraction is the operation of finding the difference between numbers. It's like taking one number away from another and is represented by the minus sign \(-\).
In our problem, once we had simplified the division, we were left with \(3 - 4\). Subtraction can sometimes yield negative results, as in this case, where removing 4 from 3 results in \(-1\).
Being the last step in this simplification exercise, subtraction solidified our final answer. Effective subtraction skills are crucial for correctly completing many arithmetic and algebraic tasks.

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Explain the difference between \(-(-3)^{2}\) and \([-(3)]^{2}\)
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