Problem 26 - Prealgebra

Question

Use the definition of exponents to expand each of the following expressions. Then multiply according to the rule for multiplication. a. \((-1)^{4}\) b. \(-1^{4}\)

Step-by-Step Solution

Answer

\((-1)^4 = 1\) and \(-1^4 = -1\).

Step 1: Understand the Definition of Exponents

Exponents mean repeated multiplication of a number. For example, \( a^n = a \times a \times \ldots \times a \) (\( n \) times). Here, you multiply the base (\( a \)) by itself \( n \) times.

Step 2: Expand Expression a

Expression \( (-1)^4 \) means multiplying \(-1\) by itself four times: \[ (-1)^4 = (-1) \times (-1) \times (-1) \times (-1) \]

Step 3: Multiply Expression a

Multiply two \(-1\)s at a time: \( (-1) \times (-1) = 1 \) and \( (-1) \times (-1) = 1 \). Then multiply the results: \( 1 \times 1 = 1 \).Hence, \( (-1)^4 = 1 \).

Step 4: Expand Expression b

Expression \(-1^4\) applies the exponent only to \( 1 \), not to \(-1\), due to the lack of brackets. So it's interpreted as: \[ - (1^4) = - (1 \times 1 \times 1 \times 1) \]

Step 5: Multiply Expression b

Calculate \( 1^4 \): \( 1 \times 1 \times 1 \times 1 = 1 \). Apply the negative sign: \( - (1) = -1 \).Hence, \(-1^4 = -1 \).

Key Concepts

Understanding Negative Numbers

Negative numbers are a common concept in mathematics, representing values less than zero. They are written with a minus sign "-" in front of them. When dealing with exponents and negative numbers, knowing how to interpret their expressions is essential. For example:

• Without brackets: \(-1^4\) means the exponent applies only to the number after the minus, so you have the expression as \(-(1^4)\). This simplifies to \(-1\).
• With brackets: \((-1)^4\) means the negative number is raised to the power in brackets, so you multiply \(-1\) by itself multiple times. Here, \((-1)\times (-1)\times (-1)\times (-1) = 1\).

Understanding the role of brackets can completely change the outcome. Brackets should be used to correctly group negative numbers when they are involved in exponentiation.

Order of Operations and Exponents

The order of operations is a set of rules to determine which calculations to perform first when evaluating an expression. In mathematics, it's commonly remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When dealing with exponents, execute them after any operations inside parentheses and before multiplication or division. In the expression \(-1^4\), neglecting the parentheses changes order: The absence of brackets applies the power only to 1. However, in \((-1)^4\), the entire \(-1\) is raised to the 4th power. This distinction emphasizes how critical parentheses are, affecting the order and the final outcome of the calculations.

Rules for Multiplication with Negative Numbers

When multiplying numbers, understanding how to handle negative numbers is crucial. Negative numbers change the sign of a product depending on how many of them are multiplied together. Here’s what happens:

• Even count of negatives: Multiplying an even number of negative numbers, like in \((-1)^4\), results in a positive number. This is because negative times negative equals positive, so four negative "-1" values multiplied together give a positive 1.
• Odd count of negatives: A product with an odd number of negative factors results in a negative. Multiplying three negative ones \((-1)^3\) would inevitably yield \(-1\).

Always remember, the sign of your product is dictated by the count: even equals positive, odd equals negative. This fundamental rule helps in interpreting results, especially when handling powers of negative numbers.