Exponents and Scientific Notation

Introduction

This section introduces the fundamental rules and properties of exponents, including the product, quotient, zero, and negative exponent rules. It also covers scientific notation, a method for expressing very large or very small numbers efficiently. Mastery of these concepts is essential for success in College Algebra and related fields.

Exponents

Exponents are a shorthand notation for repeated multiplication of the same factor. They are used extensively in algebra to simplify expressions and solve equations.

• Definition: For any real number a and natural number n, an means multiplying a by itself n times.

• Base: The number being multiplied.

• 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.

Example:

The Product Rule for Exponents

The product rule allows you to simplify expressions where the same base is multiplied with different exponents.

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

• Application: Add the exponents when multiplying like bases.

Example:

Zero Exponent Rule

Any nonzero number raised to the zero power equals one.

• Rule: If a ≠ 0, then

Example:

The Quotient Rule for Exponents

The quotient rule is used when dividing like bases with different exponents.

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

• Application: Subtract the exponent in the denominator from the exponent in the numerator.

Example:

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent.

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

• Application: Move the base with a negative exponent to the denominator and make the exponent positive.

Example:

Scientific Notation

Scientific notation is a method for writing very large or very small numbers as a product of a number between 1 and 10 and a power of 10.

• Form: where and is an integer.

• Purpose: Used in science and engineering to simplify calculations and representation of extreme values.

Steps to Write in Scientific Notation:

1. Move the decimal point in the original number until the new number is between 1 and 10.

2. Count the number of places the decimal was moved; this is the exponent on 10.

3. If the original number is greater than 10, the exponent is positive. If less than 1, the exponent is negative.

Examples:

• 4700: Move decimal 3 places left →

• 0.00047: Move decimal 4 places right →

Converting Scientific Notation to Standard Form:

• Move the decimal point the number of places indicated by the exponent.

• Positive exponent: move right; negative exponent: move left.

Examples: