Teaching Notes 7.1: Simplifying Radicals
To simplify radicals, students must identify perfect square factors, write a factorization, and take
the square root of each factor. Lack of mastery of any of these skills leads to errors in
simplification.- Explain to your students that a radical is the indicated root of a quantity.
- Explain that the opposite of squaring a number is finding its square root. For example,
102 =100, therefore
√
102 =
√
100 =10. Although both 10^2 =100 and (−10)^2 =100,
the principal square root is the positive value, which is the value that is primarily used.
The expression√
100 is called a ‘‘radical,’’ and 100, the number inside the radical symbol, is
called the ‘‘radicand.’’- Note that if the radicand is a perfect square, simplifying the radical is easy, provided students
are able to identify perfect squares. For example, because 36 is a perfect square,
√
36 =6.
- Explain that to simplify a radical when the radicand is not a perfect square, the radicand
should be factored, if possible. No factor should have a perfect square (other than 1) as one
of its factors. Note that there are several ways to begin the factoring process but the result
will always be the same. Emphasize that radicals are in simplest form if 1 is the only perfect
square factor. - Review the information and example on the worksheet with your students. Note that in the
example
√
108 is simplified two ways. The first uses the largest perfect square factor and the
second uses other perfect square factors.EXTRA HELP:
Try to find the largest perfect square factor and use this to write the factorization.ANSWER KEY:
(1) 5√
6 (2) 3
√
7 (3) 4
√
6 (4) 10
√
2 (5) 2
√
19 (6) 4
√
3
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(7)Cannot be simplified (8) 6√
10 (9)Cannot be simplified (10) 3√
6 (11) 5
√
22 (12) 6
√
7
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(Challenge)Suresh’s answer is partially correct. He has to simplify√
20 further because the
factorsof20are4and5.4isaperfectsquare.√
80 = 4
√
5
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252 THE ALGEBRA TEACHER’S GUIDE