Teaching Notes 5.6: Using Powers of Monomials
Students need to understand the two methods for simplifying the powers of monomials: finding
the power of a power or finding the power of a product. Many students confuse the steps of each.- Provide these two expressions as examples: (x^2 )^3 and (5x^2 )^3. Highlight the following points:
- Becausex^2 is a power, (x^2 )^3 is a power of a power. The exponent 3 means thatx^2 is a factor
3times.(x^2 )^3 canbewrittenasx^2 ·x^2 ·x^2 orx^6. - Because 5x^2 is the product of 5 andx^2 ,(5x^2 )^3 is the power of a product. The exponent 3
means that 5x^2 is a factor 3 times. (5x^2 )^3 can be written as 5x^2 · 5 x^2 · 5 x^2 or 125x^6.
- Becausex^2 is a power, (x^2 )^3 is a power of a power. The exponent 3 means thatx^2 is a factor
- Review the properties of integers and powers with your students.
- A negative integer raised to an even power is positive. (−2)^2 = 4
- A negative integer raised to an odd power is negative. (−2)^3 =− 8
- A positive integer raised to any power is positive. 2^3 = 8
- Review the information and examples on theworksheet with your students. Be sure to thor-
oughly discuss the property of exponents for finding the power of a power and for finding
the power of a product. If necessary, review 5.5: ‘‘Multiplying Monomials.’’
EXTRA HELP:
Express all powers of numerical coefficients as integers.ANSWER KEY:
(1)a^8 (2) 16 a^8 (3)a^16 (4) 225 x^10 (5)x^3 y^3 (6) 81 x^8 (7) 64 a^6
(8)− 27 x^3 y^6 (9) 16 x^12 y^8 (10) 16 a^2 b^2 (11) 16 a^2 b^2 (12) 1000 x^9 y^9
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(Challenge)A negative number raised to an odd power is never positive.
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