A radical is also in simplest form when the radicand 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.
Example 2. Extracting the square root. 18 has the square factor 9.
18 = 9· 2.
Therefore, is not in its simplest form. We have,
We may now extract, or 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
We will prove that when we come to rational exponents, Lesson 29. 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.
Example 3 Simplify .
Solution. = = 5.
75 has the square factor 25. And the square root of 25 times 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
= 2· 3 = 6.
Problem 1. To simplify a radical, why do we look for square factors?
To see the answer, pass your mouse over the colored area.
In order to take its square root out of the radical.
Problem 3. Simplify the following. Do that by inspecting each radicand for a square factor: 4, 9, 16, 25, and so on.
b) = = = 5
c) = = = 3
d) = = 7
e) = = 4
f) = = 10
g) = = 5
h) = = 4
Problem 4. Reduce to lowest terms.
Similar radicals have the same radicand. We add them as like terms.
2 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
Problem 6. Simplify the following.
Compare Example 4 here.
To see that 2 was a factor of the radical, you first have to simplify the radical. Compare Problem 4.
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