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.
Terms introduced in this lesson:
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.
Write in exponent form.
Encourage proper use of mathematical language:
34 is read as “three to the fourth power,” “three to the fourth,” or “three raised to the power of four.” The exponents 2 and 3 are special: 32, “three squared” and 33, “three cubed.”
(3x)3 is read as “quantity 3x cubed.”
−32 is read as “opposite of three squared.”
(−3)2 is read as “negative three squared”.
To check for conceptual understanding, ask students to translate the following into symbols.
- Square negative two. Answer: (−2)2
- Negative two squared. Answer: (−2)2
- The opposite of two squared. Answer: −22
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.
Simplify the following expressions.
Hint: Apply the exponent before multiplying.
Hint: Simplify in the parentheses first.
Hint: Simplify each term first.
Point out in Example 8 where the commutative property of multiplication is being used.
Use Examples 4 and 5 to point out two common errors:
- Multiplying bases incorrectly (exponents are different). 24⋅23≠47.
However, 34⋅24=64 because 34⋅24=(3⋅2)4=64 (exponents are same).
- Applying the product rule incorrectly (bases are different). 34⋅23≠67.
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 −2y4 does not apply to −2. Therefore, −2y4 is simplified.
a. Explain the difference between 4x2 and (4x)2.
b. Explain the difference between −12 and (−1)2.
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.
(−1)2 often gets inputted incorrectly as −1∧2.
(23)2 can get inputted incorrectly as 2/3∧2.