The objective of day 3, is to simplify radicals. WE SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when the radical has no square factors.
A radical is also in simplest form when the radical is not a fraction.
Example 1. 33, for example, has no square factors. Its factors are 3· 11, neither of which is a square number. Therefore, 
is in its simplest form.
is in its simplest form.
Example 2. Extracting the square root. 18 has the square factor 9.
18 = 9· 2.
Therefore, 
is not in its simplest form. We have,
is not in its simplest form. We have, = 
We may now take out, the square root of 9:
= = 3
. is now simplified. The radicand no longer has any square factors.
The justification for taking out the square root of 9, is this theorem:
The square root of a product
is equal to the product of the square roots
of each factor.
is equal to the product of the square roots
of each factor.
| Here is a simple illustration: |  | ) |
As for , then, it is equal to the square root of 9 times the square root of 2, which is irrational. 3.
, then, it is equal to the square root of 9 times the square root of 2, which is irrational. 3.
Example 3 Simplify 
.
.
Solution. = 
= 5
.
= = 5.
75 has the square factor 25. And the square root of 25 times 3
is equal to the square root of 25 times the square root of 3.
is equal to the square root of 25 times the square root of 3.
is now simplified.
Example 4. Simplify 
.
.
Solution. We have to factor 42 and see if it has any square factors. We can begin the factoring in any way. For example,
42 = 6· 7
We can continue to factor 6 as 2· 3, but we cannot continue to factor 7, because 7 is a prime number (Lesson 32 of Arithmetic). Therefore,
42 = 2· 3· 7
We now see that 42 has no square factors -- because no factor is repeated. Compare Example 1 and Problem 2 of the previous lesson.
therefore is in its simplest form.
Example 5. Simplify 
.
.
Solution. We must look for square factors, which will be factors that are repeated.
180 = 2· 90 = 2· 2· 45 = 2· 2· 9· 5 = 2· 2· 3· 3· 5
Therefore,
= 2· 3
= 6.
Problem 1. To simplify a radical, why do we look for square factors?
| Problem 2. Which is correct? |  |
Problem 3. Simplify the following. Do that by inspecting each radicand for a square factor: 4, 9, 16, 25, and so on.
a) 
= 
=
b) 
= 
= 
= 5
= = = 5
c) 
= 
= 
= 3
= = = 3
d) 
= 
= 7
= = 7
e) 
= 
= 4
= = 4
f) 
= 
= 10
= = 10
g) 
= 
= 5
= = 5
h) 
= 
= 4
= = 4
Problem 4. Reduce to lowest terms.
| a) |  2 | = |  2 | = |  2 | = | |
| b) |  3 | = |  3 | = |  3 | = | 2 |
| c) |  2 | = | The radical is in its simplest form. The fraction cannot be reduced. |
7 + 2 + 5 + 6 − | = | 7 + 2 + 6 + 5 − |
| = | 7 + 8 + 4. | |
2 and 6 are similar, as are 5 and −. We combine them by adding their coefficients.
and 6 are similar, as are 5 and −. We combine them by adding their coefficients.
In practice, it is not necessary to change the order of the terms. The student should simply see which radicals have the same radicand.
As for 7, it does not "belong" to any radical.
Problem 5. Simplify each radical, then add the similar radicals.
a) + 
= 3 + 2 = 5
+ = 3 + 2 = 5 b) 4 − 2 + | = | 4 − 2 + |
| = | 4· 5 − 2· 7 + | |
| = | 20 − 14 + | |
| = | 7 | |
c) 3 +  − 2 | = | 3 +  − 2 |
| = | 3· 2 + 2 − 2· 4 | |
| = | 6 + 2 − 8 | |
| = | 2 − 2 | |
d) 3 +  +  | = | 3 +  +  |
| = | 3 + 2 + 3 | |
| = | 3 + 5 | |
e) 1 −  + | = | 1 −  +  |
| = | 1 − 8 + 3 | |
| = | 1 − 5 | |
Problem 6. Simplify the following.
| a) |  2 | = |  2 | = | 2 − , | on dividing every term in the numerator by 2. |
To see that 2 was a factor of the radical, you first have to simplify the radical. Compare Problem 4.
| b) |  5 | = |  5 | = | 2 + |
| c) |  6 | = |  6 | = |  3 | on dividing every term by 2. |
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