Ray's New Elementary Algebra · Arts. 198–199 · Unit 6: Chapter VI: Powers, roots and radicals
42. Radicals and reducing them
Goal: Your student writes radicals with fractional exponents, simplifies them, and moves a coefficient under the radical sign.
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
Radicals of the second degree defined (fractional exponent 1/2, coefficients, similar radicals), then reduction to simplest form, fractional radicals, and putting a quantity under the radical sign. Answers are printed.
The lesson, step by step
Warm-up
Ask: Which of these are perfect squares: 4, 12, 25, 50, 4a², 5a²?
Radicals
Read Art. 198.
Radicals of the Second Degree are quantities affected by a square root sign whose root can not be exactly found
Say√a can also be written a with the exponent 1/2.
The principle
Read Art. 199.
The square root of the product of two or more factors is equal to the product of the square roots of those factors.
Example: √12 = √4 × √3 = 2√3.
The rule
1. Separate the quantity into two parts, one of which shall contain all the factors that are perfect squares.
2. Extract the square root of the part that is a perfect square, and prefix it as a coefficient to the other part placed under the radical sign.
Work the book's third example: √(27a³c⁴) = 3ac²√(3a).
Guided practice
Your student does exercises 1 to 3 while you watch, looking for factors 4, 9, 16, 25, 36, 49 and so on.
Fractions and the reverse
Read example 11: √(2/3) = √(6/9) = (1/3)√6. Then exercises 18 to 21: put 5, 2a and 3√5 entirely under the sign (√25, √(4a²), √45).
Independent practice
The remaining exercises 4 to 21.
Wrap-up
Ask: Why is 2√3 the "simplest form" of √12?
Tip
A list of perfect squares up to 144 on the table saves a lot of hunting.
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 ↗.