27 SIMPLIFYING RADICALSSimplifying the square roots of powers WE SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when the radicand has no square factors. 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, Example 2. Extracting the square root. 18 has the square factor 9. 18 = 9 · 2. Therefore,
We may now extract, or take out, the square root of 9:
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
Example 3 Simplify Solution. 75 has the square factor 25. And the square root of (25 times 3)
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. Therefore, 42 = 2 · 3 · 7 We now see that, because no factor is repeated, 42 has no square factors. Compare Example 1 and Problem 2 of the previous Lesson. 2 · 3 · 7 is the prime factorization of 42. 2, 3, and 7 are prime numbers.
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,
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. a) b) c) d) e) f) g) h) Problem 4. Reduce to lowest terms.
Similar radicals Similar radicals have the same radicand. We add them as like terms.
2 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)
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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