Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. In the expression \(5^{-1},\) the exponent is a _____ integer.

Step-by-Step Solution

Answer

In the expression, the exponent is a negative integer.

Step 1: Identifying the exponent

In the expression given, \(5^{-1}\), the exponent is \(-1\). We need to determine the nature of this integer.

Step 2: Classifying the integer as positive or negative

Examine the integer \(-1\). An integer is classified as negative if it has a negative sign before it. Since our integer is \(-1\), it is indeed a negative integer.

Key Concepts

Understanding Integers

Integers are a fundamental part of mathematics and consist of three subsets: positive integers, negative integers, and zero. They form the set of whole numbers that can be represented without fractional or decimal components.

Positive integers are numbers greater than zero, such as 1, 2, and 3. Zero itself is a neutral integer and is neither positive nor negative. Negative integers, on the other hand, are less than zero and often written with a minus sign, like -1, -2, and -3.

It's important to grasp that integers are used in various real-world contexts, such as in counting, measuring temperatures, and financial calculations where both deficits and surpluses are involved. Knowing how to classify and work with integers is a crucial skill in mathematics and its applications.

Exploring Negative Integers

Negative integers are an essential component of the integer set. They are the numbers that lie to the left of zero on a number line.

• These numbers are very useful for describing quantities that are typically taken away or owed, such as in financial debts or decreases in temperature.
• Negative integers can be used to express distances below a starting point, such as levels below sea level.
• A negative integer is denoted by a minus (-) sign, such as in the number -1.

When performing mathematical operations, it's vital to remember the rules specific to them. For example, when subtracting one negative integer from another, we actually perform addition. Understanding these rules helps in manipulating mathematical expressions efficiently.

Properties of Exponents

Exponents are powerful mathematical tools that express repeated multiplication of a number by itself. When dealing with exponents, there are key properties to understand, especially when negative exponents come into play.

• Product of Powers: When multiplying like bases, you add the exponents. For example, \(x^a imes x^b = x^{a+b}\).
• Quotient of Powers: When dividing like bases, you subtract the exponents: \(\frac{x^a}{x^b} = x^{a-b}\).
• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base raised to the positive exponent, such as \(x^{-n} = \frac{1}{x^n}\).
• Zero Exponent Rule: Any number raised to the zero power is 1, such as \(x^0 = 1\).

These properties are crucial for simplifying expressions and solving equations that involve powers. They become especially handy when working through algebraic problems where negative exponents appear.