Simplifying Square Roots Notes WE SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when the radicand, the number under the square root symbol, has no square factors. A radical is also in simplest form when the radicand is not a fraction. Example 1. Simplify 33 The factors of 33 are 33 and 1, 3 and 11 (33 = 3⋅11) , none of which is a square number (besides 1.which doesn’t help with simplifying)… Hence, 33 has no square factors. Therefore, form. Example 2. Simplify 33 is in its simplest 18 (Extracting the square root.) Factors of 18, are 18 and 1, 9 and 2; we notice 18 has the square factor 9. So, 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2 Note: the square root of 9 is 3 and the square root of 2 is irrational and can not be simplified any further. 18 = 3 2 is now simplified. The radicand no longer has any square factors. ----------------Thought of another way: follows that, x 2 = x because squares and square roots are opposite operations. It x2 = x ⋅ x = x . So... 1) Write a number as a product of its primes. 2) Every time you have a matching pair of prime factors, write one of them in front of the radical (or square root) symbol. Example 2 again: Simplify 18 The prime factors of 18, are 3, 3 and 2 or 18 = 32 ⋅ 2 18 = 3⋅ 3⋅ 2 = 3 2 OR 18 = 32 ⋅ 2 = 3 2 ADAPTED FROM http://www.themathpage.com/alg/simplify-radicals.htm Example 3. Simplify 75 . 75 = 25 ⋅ 3 = 5 3 Solution. OR 75 = 5 ⋅ 5 ⋅ 3 Example 4. Simplify (or ) 52 ⋅ 3 = 5 3 42 . Solution. 42 = 6 ⋅ 7 = 2 ⋅ 3⋅ 7 We now see that 42 has no square factors— because no factor is repeated. Therefore simplest form. Example 5. Simplify 42 is in its 180 . Solution. 180 = 10 ⋅18 = 2 ⋅ 5 ⋅ 2 ⋅ 9 = 2 ⋅ 5 ⋅ 2 ⋅ 3⋅ 3 = 2 2 ⋅ 32 ⋅ 5 = 2 ⋅ 3 5 = 6 5 Example 6. Simplify 27 . Solution. 27 = 3⋅ 3⋅ 3 = 32 ⋅ 3 = 3 3 Similar radicals Similar radicals have the same radicand. We add (combine) them as like terms. 7+2 3+5 2 +6 3− 2 = 7+4 2 +8 3 Notice: 2 3 and 6 3 are like terms and 5 2 and − 2 are like terms. We combine them by adding their coefficients. ADAPTED FROM http://www.themathpage.com/alg/simplify-radicals.htm Simplifying Square Roots Name__________________________________ Date__________________ Class ___________ 1) Which is correct? 7 ⋅ 4 = 7 4 or 2 7 2) Simplify the following. (Do that by inspecting each radicand for a square factor: 4, 9, 16, 25, and so on.) a) 28 = b) 50 = c) 45 = d) 98 = e) 48 = f) 300 = g) 150 = h) i) 80 = 125 = 3) Reduce to lowest terms. 20 = a) 2 b) 72 = 3 c) 22 = 2 d) 300 = 5 e) 98 = 14 28 = 14 4) Simplify each radical, then add the similar radicals. f) a) 18 + 8 = ADAPTED FROM http://www.themathpage.com/alg/simplify-radicals.htm b) 4 75 − 2 147 + 3 = c) 3 28 + 88 − 2 112 = d) 3 + 24 + 54 = e) 1− 128 + 18 = 5) Simplify the following. (Hint: Use the Distributive Property to divide each term in the numerator by the denominator (or a common factor of the denominator).) a) 4− 8 = 2 b) 10 + 50 = 5 c) 6 + 24 = 6 d) 18 − 8 + 6 = 6 ADAPTED FROM http://www.themathpage.com/alg/simplify-radicals.htm

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