8.1: Exponent Properties Involving Products
Learning Objectives
At the end of this lesson, students will be able to:
- Use the product of a power property.
- Use the power of a product property.
- Simplify expressions involving product properties of exponents.
Vocabulary
Terms introduced in this lesson:
- power
- exponent
- square, cube
- base
- factors of the base
- product rule for exponents
- power of a product
- power rule for exponents
Teaching Strategies and Tips
Encourage students to review basic exponents in the chapter Real Numbers.
- An exponent is a notation for repeated multiplication of a number, variable, or expression.
- Exponents count how many bases there are in a product
- Parentheses precede exponents in the order of operations.
Additional Examples:
Write in exponent form.
a.
b.
c.
d.
Encourage proper use of mathematical language:
- is read as “three to the fourth power,” “three to the fourth,” or “three raised to the power of four.” The exponents and are special: , “three squared” and , “three cubed.”
- is read as “quantity cubed.”
- is read as “opposite of three squared.”
- is read as “negative three squared”.
To check for conceptual understanding, ask students to translate the following into symbols.
- Square negative two. Answer:
- Negative two squared. Answer:
- The opposite of two squared. Answer:
Allow the class to infer the product rule for exponents in Example 2.
- “When you multiply like bases you add the exponents.”
Allow the class to infer the power rule for exponents in Example 6c.
- “When you raise an exponent to an exponent, you multiply them.”
Combine exponent rules in Examples 6-9.
- Suggest that students apply each rule one step at a time.
- Order of operations must be followed at each step: evaluate exponents before multiplying. See Examples 6a, 6b, and 9.
Additional Examples:
Simplify the following expressions.
a.
Hint: Apply the exponent before multiplying.
b.
Hint: Simplify in the parentheses first.
c.
Hint: Simplify each term first.
Point out in Example 8 where the commutative property of multiplication is being used.
Error Troubleshooting
Use Examples 4 and 5 to point out two common errors:
- Multiplying bases incorrectly (exponents are different). .
- However, because (exponents are same).
- Applying the product rule incorrectly (bases are different). .
Remind students in Review Question 6 that even powers of negative numbers are positive. Try using a visual to show that negatives cancel in pairs:
General Tip: Occasionally, students forget that:
In Review Question 13, remind students that the exponent in does not apply to . Therefore, is simplified.
Additional Examples:
a. Explain the difference between and .
b. Explain the difference between and .
Teachers are encouraged to survey the class for answers. Ask: What constitutes the base of an exponent?
General Tip: Encourage students to think about syntax before inputting an expression into a calculator.
- often gets inputted incorrectly as .
- can get inputted incorrectly as .