Ray's New Elementary Algebra · Arts. 191–194 · Unit 6: Chapter VI: Powers, roots and radicals
40. Roots of fractions and decimals; approximate roots
Goal: Your student finds square roots of fractions and decimals, and approximate roots of numbers that are not perfect squares.
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
Square roots of fractions; perfect and imperfect squares and surds, with the proof that the root of an imperfect square is not a fraction; approximate roots to within a fraction or to so many decimals. Answers are printed.
The lesson, step by step
Warm-up
Ask: What is the square root of 4/9? Of 0.25?
Fractions
Read Art. 191.
The square root of a fraction is found by extracting the square root of both terms.
Example: 12/27 reduces to 4/9 first, whose root is 2/3.
Perfect squares and surds
Read Art. 192.
A Perfect Square is a number whose square root can be exactly ascertained; as, 4, 9, 16, etc.
A Surd is a root which can not be exactly expressed.
Read the proof on pp. 168 and 169 together. It is short and beautiful; take it slowly.
Approximate roots
Read Art. 193 and its rule. Work the book's example: the root of 2 to within 1/3. 2 × 9 = 18; the root of 18 is about 4; so the answer is 4/3.
Decimals
Read the rule for decimals: add two ciphers (zeros) for each decimal place wanted. Work together:
6. Find the square root of 2 to six places of decimals.
(1.414213.) Then exercise 7 of Art. 194, the square root of 1.1025 (1.05).
Independent practice
Art. 191 exercises 1 to 3, Art. 193 exercises 1 to 5 and 7, and Art. 194 exercises 4 and 5. Use the site's calculator (/homeschool/calculator/) to check a decimal root only after working it.
Wrap-up
Ask: Why can't the square root of 2 ever be written exactly as a fraction?
Tip
Pair the decimal digits in twos from the decimal point, both ways. Wrong pairing is the usual mistake.
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 ↗.