Exponential Equations and Inequalities

Review: Exponent Properties

• Before we get to the main topic, the most basic laws of exponents are most heavily used in this and the following topics.
• It is important that we review them before we proceed.

The Definition of Exponentiation

• Exponentiation is the process of repeated multiplication.

  • It is analogous to how multiplication is repeated addition.

• It has three parts:

  • the base (cyan): the number to be exponentiated.
  • the exponent (orange): how many times to multiply
  • the power (magenta): the answer in exponentiation

• To compute an exponent, we simply multiply the base to itself repeatedly.
  • The amount of times to multiply is based on the exponent.

Worked Example
Evaluating Exponents

Evaluate the following exponents

Solution

We just repeatedly multiply the base to itself based on the value of the exponent and we should be done.

In general, if we have an expression raised to some exponent , it should be equal to:

Multiplication, Division, and Powers of Powers

• Now we discuss the three most basic properties of exponents.
• If we multiply two powers with the same base, it should be the same as if we are adding their exponents.

• If we divide two powers with the same base, it is the same as if we are subtracting their exponents.

• If we raise a power to an exponent, it is the same as multiplying the exponents together.

• We try to use these properties in action by simplifying expressions.

Worked Example
Simplifying Algebraic Expressions with Exponents

Simplify the following expressions

Solution

For , we first use the distributive property, then we apply the product rule.

For , we first use the quotient rule, then the power rule.