Laws of exponents
Exponential notation is a short way of writing the same number multiplied by itself many times. We will now have a closer look at writing numbers using exponential notation. Exponents can also be called indices.
For any real number and natural number , we can write a multiplied by itself times as .
Identity 1

( because is undefined)

( because is undefined)
Examples:
Notice that we always write the final answer with positive exponents.
Chapter introduction
Laws of exponents
There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been done in earlier grades, but we list all the laws here for easy reference:
Identity 2
where , and .
Example 1: Applying the exponential laws
Question
Simplify:
Answer
Example 2: Exponential expressions
Question
Simplify:
Answer
Change the bases to prime numbers
Simplify the exponents
Example 3: Exponential expressions
Question
Simplify:
Answer
Change the bases to prime numbers
Subtract the exponents (same base)
Write the answer as a fraction
Important: when working with exponents, all the laws of operation for algebra apply.
Example 4: Simplifying by taking out a common factor
Question
Simplify:
Answer
Simplify to a form that can be factorised
Take out a common factor
Cancel the common factor and simplify
Example 5: Simplifying using difference of two squares
Question
Simplify:
Answer
Change the bases to prime numbers
Factorise using the difference of squares
Simplify
Exercise 1:
Simplify without using a calculator:
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