Ray's New Elementary Algebra · Arts. 178–182 · Unit 6: Chapter VI: Powers, roots and radicals
37. Powers of monomials, polynomials and fractions
Goal: Your student raises monomials, polynomials and fractions to powers, with the right signs.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Involution: power, root and exponent defined; Case I (monomials) with the sign rule and fifteen exercises; Case II (polynomials by repeated multiplication); Case III (fractions). Answers are printed.
The lesson, step by step
Warm-up
Ask: What is (−2)², (−2)³, (−2)⁴?
Monomials
Read Art. 179: (2ab²)³ = 8a³b⁶. Then the rule:
1. Raise the numeral coefficient to the required power, and multiply the exponent of each letter by the exponent of the power.
2. If the monomial is positive, all the powers will be positive; if negative, the even powers will be positive, and the odd powers negative.
Guided practice
Your student does exercises 3, 4 and 12 while you watch, saying the sign rule aloud each time. Compare with the printed answers.
Polynomials
Read Art. 181.
Find the product of the quantity, taken as a factor as many times as there are units in the exponent of the power.
Work together: the cube of a + x is a³ + 3a²x + 3ax² + x³.
Fractions
Read Art. 182.
Raise both numerator and denominator to the required power.
Independent practice
Case I exercises 1 to 15 (or every other one), Case II exercises 2 to 7, and Case III exercises 1 to 4. After working by hand, check one polynomial on the site's calculator (/homeschool/calculator/) with expand((2*x-1)^3).
Wrap-up
Ask: Why is (−a)⁶ positive but (−a)⁵ negative?
Tip
Multiplying exponents (not adding) is right here, because a power of a power repeats the whole factor.
Answers to the book's problems
The book prints the answer after each exercise.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.