Exponents, Polynomials, and Polynomial Functions

Exponents and Their Properties

Exponents are a mathematical shorthand for repeated multiplication of a base number. Understanding the rules for manipulating exponents is essential for simplifying algebraic expressions and solving equations.

• Base: The number being multiplied repeatedly.

• Exponent (Power): Indicates how many times the base is used as a factor.

• Order of Operations: Exponents are evaluated before multiplication, division, addition, or subtraction.

Product Rule for Exponents

The Product Rule allows us to simplify expressions where the same base is raised to different exponents and multiplied together.

• Rule: For any real number a and integers m and n:

• Example:

Zero Exponent Rule

Any nonzero number raised to the zero power equals one. This rule is fundamental for simplifying expressions and understanding the structure of exponents.

• Rule: For any nonzero real number a:

• Example:

Quotient Rule for Exponents

The Quotient Rule is used when dividing like bases with exponents. It simplifies the expression by subtracting the exponents.

• Rule: For any nonzero real number a and integers m and n:

• Example:

Negative Exponents

Negative exponents indicate reciprocals. A base raised to a negative exponent equals one divided by the base raised to the corresponding positive exponent.

• Rule: For any nonzero real number a and positive integer n:

• Example:

• Note: Parentheses affect the base. For example, but .

Writing with Positive Exponents

Expressions with negative exponents should be rewritten using positive exponents for clarity and standard form.

• Example:

• Example:

Scientific Notation

Definition and Purpose

Scientific notation is a method for expressing very large or very small numbers in the form , where and is an integer. This notation is widely used in science and engineering for clarity and efficiency.

Converting to Scientific Notation

• Move the decimal point in the original number to create a new number between 1 and 10.

• Count the number of places the decimal was moved:

  • If the original number is greater than or equal to 10, the exponent is positive.

  • If the original number is less than 1, the exponent is negative.

• Write the number as the product of the new value and raised to the counted exponent.

Example:

• 4700 → Move decimal 3 places left:

• 0.00047 → Move decimal 4 places right:

Converting from Scientific Notation to Standard Notation

• If the exponent is positive, move the decimal point to the right.

• If the exponent is negative, move the decimal point to the left.

Example:

• (move decimal 3 places right)

• (move decimal 5 places left)