Exponent

Definition

An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself. It is a concise way to express repeated multiplication of the same number.

5 Must Know Facts For Your Next Test

1. Exponents are used to simplify expressions involving repeated multiplication of the same number.
2. The laws of exponents, such as $a^m \cdot a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$, allow for efficient manipulation of exponential expressions.
3. Rational exponents, such as $a^{\frac{1}{2}}$, represent the square root of the base number.
4. Exponential functions, where the independent variable is the exponent, exhibit unique growth and decay patterns.
5. The properties of logarithms are closely related to the properties of exponents, allowing for the evaluation and simplification of exponential expressions.

Review Questions

• Explain how exponents are used in the context of adding and subtracting polynomials.
  • Exponents play a crucial role in the addition and subtraction of polynomials. When combining like terms, the exponents of the variable(s) must be the same for the terms to be combined. For example, in the expression $3x^2 + 2x^2$, the exponent of $x$ is 2 for both terms, allowing them to be combined into $5x^2$. Proper understanding of exponents is essential for simplifying polynomial expressions through addition and subtraction.
• Describe the properties of exponents and how they are used in scientific notation.
  • The properties of exponents, such as $a^m \cdot a^n = a^{m+n}$ and $a^{\frac{1}{n}} = \sqrt[n]{a}$, allow for efficient manipulation of exponential expressions. These properties are particularly useful in the context of scientific notation, where very large or very small numbers are expressed as a product of a number between 1 and 10 and a power of 10. Mastering the properties of exponents is crucial for working with and converting between standard and scientific notation.
• Explain how exponents are used in the simplification of rational exponents and the solving of radical equations.
  • Rational exponents, such as $a^{\frac{1}{2}}$, represent the square root of the base number $a$. The laws of exponents can be extended to handle rational exponents, allowing for the simplification of expressions involving fractional powers. Furthermore, the relationship between rational exponents and radicals is used to solve radical equations, where the goal is to isolate the variable and express the solution in terms of a rational exponent. Proficiency in working with rational exponents is essential for simplifying and solving a variety of algebraic expressions and equations.

Related terms

Base: The number that is being multiplied by itself, as indicated by the exponent.

Power: Another term for exponent, referring to the number that represents how many times the base is multiplied by itself.

Scientific Notation: A way of writing very large or very small numbers using exponents to indicate the order of magnitude.