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Chapter 1: Problem 119

Explain why \(-3^{2}\) and \((-3)^{2}\) simplify to different numbers.

Short Answer

Expert verified
\(-3^{2} = -9\) and \((-3)^{2} = 9\); parentheses affect the order of operations.

Step by step solution

01

Understanding the Expression -3虏

The expression \(-3^{2}\) involves the exponentiation of the number 3 while maintaining the negative sign outside the exponentiation. It must be interpreted as \(-(3^{2})\), which means we first calculate \(3^{2}\) and then apply the negative sign.
02

Calculating 3虏

Calculate \(3^{2}\), which means multiplying 3 by itself. \[ 3^{2} = 3 imes 3 = 9 \]
03

Applying the Negative Sign

After finding \(3^{2} = 9\), apply the negative sign in front to the result. \[ -3^{2} = -(9) = -9 \]
04

Understanding the Expression (-3)虏

In the expression \((-3)^{2}\), the negative sign is within the parentheses, meaning the entire term \(-3\) is being squared.
05

Calculating (-3)虏

Calculate \((-3)^{2}\), which involves multiplying \(-3\) by itself. \[ (-3)^{2} = -3 imes -3 = 9 \]
06

Comparing Results

From the calculations, \(-3^{2}\) yields -9, while \((-3)^{2}\) yields 9. The parentheses change the operation, squaring the entire term including the negative sign, resulting in different outcomes.

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Key Concepts

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

Negative Numbers
Negative numbers are numbers less than zero, represented by a minus sign (-). They are essential in mathematics, especially when dealing with temperatures, debts, or even the opposite direction in physics.
When performing mathematical operations with negative numbers, consider:
  • A negative sign before a number indicates it is less than zero, such as -5.
  • Multiplying two negative numbers results in a positive number, such as \(-2 \times -3 = 6\).
  • Adding a negative number means subtracting its absolute value, like \(5 + (-3) = 5 - 3 = 2\).
This understanding helps when dealing with expressions like \(-3^{2}\) and \((-3)^{2}\). Notice how the negative sign affects the result of calculations through placement and grouping.
Parentheses in Expressions
Parentheses play a crucial role in mathematics by determining which operations are performed first. They are used to group numbers and operations, ensuring correct interpretation and calculation.
Consider expressions with and without parentheses:
  • Without parentheses: In the expression \(-3^{2}\), the exponent only applies to 3, not the negative sign, because the calculation follows as \-1 \cdot (3^{2})\.
  • With parentheses: In the expression \((-3)^{2}\), both the negative number and the number 3 are squared together, as the parentheses group them as one entity, leading to \((-3) \cdot (-3)\).
Using parentheses wisely can alter the result significantly, which highlights their importance in mathematical expressions.
Order of Operations
The order of operations is a fundamental principle that guides which operations to perform first in a mathematical expression. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it ensures consistent and correct results.
  • Parentheses: Perform the operations inside parentheses first.
  • Exponents: Next, solve any exponents.
  • Multiplication and Division: These operations are done from left to right.
  • Addition and Subtraction: Finally, perform these operations from left to right.
This order is crucial in solving expressions like \(-3^{2}\) and \((-3)^{2}\). Apply the order of operations by first addressing parentheses, then moving to exponents, and lastly addressing any multiplications involving negative numbers.

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Most popular questions from this chapter

In your own words, explain why the empty set is a subset of every set.

Simplify each expression. See Examples I through 11 . $$ \frac{(3-\sqrt{9})-(-5-1.3)}{-3} $$

The boxed definition of absolute value states that \(|a|=-a\) if \(a\) is a negative number. Explain why \(|a|\) is always non negative, even though \(|a|=-a\) for negative values of \(a\)

Name the only real number that is its own opposite, and explain why this is so.

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ \frac{\sqrt{x}}{y}-\frac{y}{x} $$
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