Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
Integer exponents are a way of expressing repeated multiplication of the same number. They are used to represent the power or degree to which a number is raised, where the exponent is a whole number, either positive or negative.
The value of a number raised to a positive integer exponent is always greater than 1, while the value of a number raised to a negative integer exponent is always less than 1.
Any number raised to the power of 0 is equal to 1, regardless of the base.
The product of two numbers raised to the same exponent is equal to the product of their bases raised to that exponent.
The quotient of two numbers raised to the same exponent is equal to the quotient of their bases raised to that exponent.
Review Questions
Explain the relationship between the base and the exponent in integer exponents, and how this affects the value of the expression.
In integer exponents, the base represents the number being multiplied, and the exponent represents the number of times the base is multiplied by itself. Positive integer exponents result in a value greater than 1, as the base is repeatedly multiplied. Negative integer exponents result in a value less than 1, as the base is repeatedly divided. The value of a number raised to the power of 0 is always 1, regardless of the base.
Describe the properties of integer exponents that allow for the simplification of expressions involving multiplication and division of numbers raised to powers.
The properties of integer exponents allow for the simplification of expressions involving multiplication and division. The product of two numbers raised to the same exponent is equal to the product of their bases raised to that exponent. The quotient of two numbers raised to the same exponent is equal to the quotient of their bases raised to that exponent. These properties make it possible to simplify complex expressions involving integer exponents by focusing on the relationships between the bases and exponents.
Analyze how the understanding of integer exponents can be applied to solve problems involving exponential growth or decay, and explain the significance of this concept in real-world scenarios.
The understanding of integer exponents is crucial in modeling and solving problems involving exponential growth or decay, which are commonly observed in various real-world scenarios. For example, the concept of integer exponents can be applied to understand the growth of bacterial populations, the depreciation of assets, the compounding of interest rates, and the spread of infectious diseases. By recognizing the patterns and properties of integer exponents, one can make accurate predictions, analyze trends, and make informed decisions in these contexts, which have significant implications for fields such as finance, economics, biology, and public health.
The number that represents the power or degree to which the base is raised.
Exponential Notation: The way of expressing repeated multiplication using a base and an exponent, such as $a^n$, where $a$ is the base and $n$ is the exponent.