1. Operations with Real Numbers
1.5. Exponents and Scientific Notation
Learning Objectives
By the end of this section it is expected that you will be able to:
- Simplify expressions with exponents
- Simplify expressions with zero exponents
- Use the definition of a negative exponent
- Use formulas with exponents in applications
- Convert from decimal notation to scientific notation
- Convert scientific notation to decimal form
- Multiply and divide using scientific notation
Simplify Expressions with Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, means to multiply 2 by itself 4 times, so means 2 · 2 · 2 · 2
Let’s review the vocabulary for expressions with exponents.
Exponential Notation (Power)
This is read to the power.
In the expression , the exponent tells us how many times we use the base as a factor.
Before we begin working with expressions containing exponents, let’s simplify a few expressions involving only numbers.
EXAMPLE 1
Simplify: a) b) c) d) .
| a) | |
| Multiply three factors of 4. | 4 · 4 · 4 |
| Simplify. | |
| b) | |
| Multiply one factor of 7. | |
| c) | |
| Multiply two factors. | |
| Simplify. | |
| d) | |
| Multiply two factors. | |
| Simplify. |
TRY IT 1
Simplify: a) b) c) d) .
Show answer
a) 216 b) c) d) 0.1849
TRY IT 2
Simplify: a) b) .
Show answer
a) b)
Notice the similarities and differences in (Example 2) a) and (Example 2) b)! Why are the answers different? As we follow the order of operations in part a) the parentheses tell us to raise the to the 4th power. In part b) we raise just the 5 to the 4th power and then take the opposite.
Simplify Expressions with an Exponent of Zero
When simplifying expressions with exponents we very often use the Product Property and the Quotient Property.
Product Property for Exponents
If is a real number, and and are counting numbers, then
To multiply with like bases, add the exponents.
An example with numbers helps to verify this property.
Quotient Property for Exponents
If is a real number, , and and are whole numbers, then
A couple of examples with numbers may help to verify this property.
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like . From your earlier work with fractions, you know that:
In words, a number divided by itself is 1. So, , for any , since any number divided by itself is 1
The Quotient Property for Exponents shows us how to simplify when > and when < by subtracting exponents. What if ?
Consider , which we know is 1
| Write as . | |
| Subtract exponents. | |
| Simplify. |
Now we will simplify in two ways to lead us to the definition of the zero exponent. In general, for :
We see simplifies to and to 1. So .
Zero Exponent
If is a non-zero number, then .
Any nonzero number raised to the zero power is 1
Simplify: .
The definition says any non-zero number raised to the zero power is 1
| Use the definition of the zero exponent. |
TRY I