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Chapter 5: Problem 6

In the expression \(5^{-1},\) the exponent is a __________ integer.

Short Answer

Expert verified
negative

Step by step solution

01

Identify the Exponent

In the expression \(5^{-1}\), the number 5 is the base and -1 is the exponent. The exponent is the number that tells us how many times to multiply the base by itself.
02

Determine the Sign of the Exponent

Look at the exponent -1 in the expression. The '-' sign in front of the 1 indicates that the exponent is negative, different from a positive integer that has no sign or a plus sign in front of it.
03

Classify the Type of Integers

Integers can be positive (e.g., 1, 2, 3), zero, or negative (e.g., -1, -2, -3). Since the exponent is -1, it falls under the category of negative integers.

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Key Concepts

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

Integer Classification
Integers are a set of numbers that include whole numbers and their negatives. They play a vital role in mathematics as the building blocks for algebra and other advanced topics. Integers can be classified into three main categories:
  • Positive Integers: These are numbers greater than zero, such as 1, 2, 3, and so on. They do not have any sign or have a plus sign (+).
  • Zero: It is a unique integer that is neither positive nor negative.
  • Negative Integers: These numbers are less than zero and have a minus sign (-) in front of them, such as -1, -2, -3, etc.
Understanding integer classification helps in identifying the nature of numbers in algebraic expressions, enabling better manipulation and comprehension.
Base and Exponent
In algebraic expressions, the terms 'base' and 'exponent' are key in expressing repeated multiplication. The base is the number that is multiplied by itself, while the exponent indicates how many times the base is to be multiplied.
  • Base: This is the number that we raise to a power. For example, in the expression \(a^n\), \(a\) is the base.
  • Exponent: It is the small number written slightly above and to the right of the base. It tells us how many times we multiply the base by itself, such as \(a^3 = a \times a \times a\).
When the exponent is negative, it signifies the reciprocal of the positive exponent. So, \(a^{-n} = \frac{1}{a^n}\). This concept is particularly useful in simplifying algebraic expressions and solving equations.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and arithmetic operations to represent mathematical ideas concisely. They can include terms with both positive and negative exponents.
  • Components: Algebraic expressions consist of terms, which are products of numbers and variables, such as \(3x^2\) and \(-5y\).
  • Operations: They include addition, subtraction, multiplication, and division. Exponents indicate repeated multiplication within these expressions.
  • Simplification: By understanding the rules of exponents and integer operations, algebraic expressions can be simplified or expanded to solve equations or evaluate expressions more easily.
An important aspect of mastering algebra is knowing how to manipulate these expressions efficiently, which includes applying rules for negative exponents in calculations.

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