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After completing this lesson, you will be able to say:
• I can write numerical expressions involving whole-number exponents.
• I can evaluate numerical expressions involving whole-number exponents.
• I can solve order of operation expressions that contain exponents.
Key Terms
• Exponential form: A number including a base and an exponent.
• Base: The number that is multiplied by itself when written in exponential form.
• Exponent:A number that is written above and to the right of a base to indicate how many times to multiply the
base by itself; sometimes called a power
Exponential form
Exponential form is just a simplified way of writing a multiplication expression where a number is being multiplied by itself
Area in Exponential Form
Since the 5 is being multiplied by itself 2 times, you can use an exponent of 2. The area 5 ft × 5 ft written in exponential form is 52 ft2.
When the exponent is a 2, this is called squaring the base. So you can say "five squared."
Volume in Exponential Form
To calculate the volume of the circus cube you would multiply 5 ft × 5 ft × 5 ft.
5 is the base, but this time it is multiplied 3 times so the exponent in this case is 3. Therefore, the exponential form of the volume is 53 ft3.
When an exponent is a 3, this is called cubing the base. So you can say "five cubed."
Example using Exponential Form
The goal of this new circus act is for the performers to knock over as many pins as possible. Each pin will knock over three other pins, and each of those will knock over three more pins, and so on.
There are five total rows of pins. The expression to see how many pins to knock down in the fifth row is created by multiplying 3 five times. You can write this expression as 3 × 3 × 3 × 3 × 3 or in exponential form as 35
Try it
Ginger, the circus mouse, gave birth to twins. Each of the twins then gave birth to twins. Then those twins gave birth to twins.
Check your work
To understand how the mice population grew, you would multiply 2 three times.
So 2 × 2 × 2 = 23 or "two cubed."
Reading Exponents
An exponent is sometimes referred to as a power. So 52 can be read as "five to the power of two."
Here are a few other variations for reading exponential expressions:
52 53 54
5 to the second power 5 to the third power 5 to the fourth power
5 to the power of 2 5 to the power of 3 5 to the power of 4
5 squared 5 cubed
5 raised to the second power 5 raised to the third power 5 raised to the fourth power
5 with an exponent of 2 5 with an exponent of 3 5 with an exponent of 4
Typing Exponents
Typing Exponents
An easy way to represent an exponent is to use the ^ symbol (above the number 6 on your keyboard).
So, 53 can be typed as 5^3.
Example: 64 = 6^4
Simplifying exponential numbers
35 = 3 x 3 x 3 x 3 x 3 = 3 x 3 x 3 x 3 x 3
9 x 3 x 3 x 3
27 x 3 x 3
81 x 3
243
When simplifying an exponent, you must remember that 73 = 7 × 7 × 7. It does not equal 7 × 3 or 7·3 or 73
Caution
Try it
Simplify the exponential expression of 6.24. Be sure to round your answer to the nearest tenths place
Check your work
6.24 = 6.2 × 6.2 × 6.2 × 6.2 = 1,477.6336
This is 1,477.6 when rounded to the tenths place
Evaluating Numerical Expressions
When simplifying an expression, you must always follow the order of operations.
Order of operations:The rules of which calculation comes first when evaluating an expression
Simplifying and Expression
Steps to Simplify an expression
Step 1: simplify inside parenthesis
Step 2: simplify the exponents
Step 3: evaluate any multiplication and/or division from left to right
Step 4: complete any addition and/or subtraction from left to right