Rules of Exponents

Introduction to Exponents

An exponential expression is any expression containing one or more exponents. In the expression , a is the base and n is the exponent or power. To evaluate an exponent, multiply the base by itself as many times as the value of the exponent. For example:

Key Point: The difference between and is that means , while .

Rules of Exponents

Exponent rules allow us to simplify expressions involving powers. The main rules are summarized below:

Product Rule

The Product Rule states that when multiplying like bases, add the exponents:

• Formula:

• Example:

Quotient Rule

The Quotient Rule states that when dividing like bases, subtract the exponents:

• Formula:

• Example:

Power Rule

The Power Rule states that when raising a power to another power, multiply the exponents:

• Formula:

• Example:

Power of a Product

The Power of a Product Rule states that a product raised to a power equals each factor raised to that power:

• Formula:

• Example:

Power of a Quotient

The Power of a Quotient Rule states that a quotient raised to a power equals the numerator and denominator each raised to that power:

• Formula:

• Example:

Zero Exponent Rule

The Zero Exponent Rule states that any nonzero base raised to the zero power is 1:

• Formula: for

• Example:

Negative Exponent Rule

The Negative Exponent Rule states that a base raised to a negative exponent equals the reciprocal of the base raised to the positive exponent:

• Formula:

• Example:

Summary Table: Rules of Exponents

Applications of Exponents

Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten. A number written in the form where is an integer greater than 0 and less than 10, and is any integer.

• To convert a number written in standard form to scientific notation:

  • Move the decimal point until the number is between 1 and 10.

  • The number of places moved is the exponent .

  • If moved right, is negative. If moved left, is positive.

• To convert from scientific notation to standard form:

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

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

Example: in standard form is .

Operations with Scientific Notation

• Multiplication: Multiply the coefficients and add the exponents.

• Division: Divide the coefficients and subtract the exponents.

Applied Problems

• Distance Example: The moon is approximately miles from Earth. In standard form, this is miles.

• Economics Example: The GDP of the United States was in 2005. In scientific notation, this is .

• Social Media Example: If Facebook users upload 250 million photos daily, the total in a year (365 days) is photos.

Compound Interest

Compound Interest Formula

The compound interest formula calculates the amount of money in an account after a certain time period:

• Formula:

• Where:

  • = amount after years

  • = principal (initial amount)

  • = annual interest rate (as a decimal)

  • = number of times compounded per year

  • = number of years

• Example: If , , , , then

Geometry and Exponents

Volume and Surface Area Formulas

• Rectangular Solid: where = length, = width, = height.

• Cylinder: where = radius, = height.

• Example: For a cylinder with ft and ft, ft

Concept Check

• If an exponent is not written, it is assumed to be 1 (e.g., ).

• In scientific notation, the standard form is where is between 1 and 10, not including 10.

Practice Problems

• Simplify using exponent rules and write answers with positive exponents.

• Examples include:

Additional info: These notes cover College Algebra topics from Chapter 5 (Exponents and Polynomials) and include applications relevant to Chapters 10 and 12 (Rational Exponents, Radicals, and Exponential Functions).