Exponents

Definition

Exponents, also known as powers, are mathematical expressions that represent repeated multiplication of a number or variable by itself. They are used to concisely express large or small quantities and to model various types of functions, including power functions and polynomial functions.

5 Must Know Facts For Your Next Test

1. Exponents can be used to represent very large or very small numbers in a compact way, making calculations and comparisons easier.
2. The value of an exponent represents the number of times the base is multiplied by itself. For example, $x^3$ means $x$ multiplied by itself three times.
3. Exponents can be positive, negative, or zero, each with different meanings and applications.
4. Exponents are a fundamental component of power functions, which are used to model a wide range of phenomena, from population growth to radioactive decay.
5. Polynomial functions, which are sums of terms with different exponents, also rely on the properties of exponents for their manipulation and analysis.

Review Questions

• Explain how exponents are used to represent large or small quantities in a compact way.
  • Exponents allow us to express very large or very small numbers in a concise manner. For example, the number $1,000,000,000$ can be written as $10^9$, which is much more compact and easier to work with. Similarly, the very small quantity $0.000001$ can be expressed as $10^{-6}$. This compact representation makes it easier to compare and perform calculations with these types of numbers.
• Describe the relationship between exponents and power functions, and how this connection is important in the context of this chapter.
  • Power functions, which are functions of the form $f(x) = x^n$, where $n$ is the exponent, are a crucial topic in this chapter. Exponents are the foundation of power functions, as they determine the rate of change and the shape of the function. Understanding the properties of exponents, such as how they affect the behavior of the function, is essential for analyzing and working with power functions, which are used to model a wide range of real-world phenomena.
• Analyze how the properties of exponents, such as positive, negative, and zero exponents, are applied in the context of polynomial functions.
  • Polynomial functions are composed of terms with different exponents. The properties of exponents, including positive, negative, and zero exponents, are crucial for manipulating and simplifying polynomial expressions. For example, the ability to apply rules like $x^a \cdot x^b = x^{a+b}$ and $x^a \div x^b = x^{a-b}$ allows us to perform operations on polynomial terms and understand their behavior. Understanding these exponent properties is essential for working with and analyzing polynomial functions, which are a key focus in this chapter.

Related terms

Base: The number or variable that is being multiplied by itself a certain number of times, as indicated by the exponent.

Exponent: The number that represents the number of times the base is multiplied by itself.

Power Function: A function of the form $f(x) = x^n$, where $n$ is the exponent, which can be used to model various real-world phenomena.