Ray's New Elementary Algebra · Arts. 204–205 · Unit 6: Chapter VI: Powers, roots and radicals
44. Rational denominators and equations with radicals
Goal: Your student clears a radical from a denominator and solves simple equations containing square roots, checking for false roots.
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
Making a denominator rational (multiply by the radical, or by the binomial with its sign changed), with exercises; then three directions for solving equations with radicals, worked examples and exercises 1 to 19. Answers are printed. A general review ends the chapter.
The lesson, step by step
Warm-up
Multiply (3 + √2)(3 − √2). (7.)
Rational denominators
Read Art. 204 and its rule:
If the denominator is a monomial, multiply both terms by it; but if it is a binomial, multiply both terms by the denominator, with the sign of the second term changed.
Your student does exercises 1 and 2 on p. 186. Read the Remark: this makes decimal values much easier to compute.
Equations with a radical
Read Art. 205 and its first direction:
1st. When the equation contains one radical expression, transpose it to one side of the equation, and the rational terms to the other, then involve both sides.
Work the book's example: √(x − 1) − 1 = 2; √(x − 1) = 3; x − 1 = 9; x = 10.
Two radicals
Follow the third example on p. 187: √(x − 5) − 3 = 4 − √(x − 12). Put one radical alone, square, simplify, square again: x = 21.
Guided practice
Your student does exercises 1 and 2 on p. 187 while you watch, and checks each answer in the original equation (13 and 5).
Independent practice
Exercises 3, 5, 6, 9, 12 and 13. Check every answer by substitution. After working by hand, your student may confirm one with solve on the site's calculator (/homeschool/calculator/).
Wrap-up
Use the General Review on p. 188 as an oral quiz on the chapter.
Tip
Squaring both sides can bring in an answer that doesn't fit the original equation. That is why checking by substitution is not optional here.
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 ↗.