yosty

Simplifying radicals or in other words simplifying square roots:) http://www.themathpage.com/alg/simplify-radicals.htm#simplest Copyright © 2001-2009 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com

What does it me to simplify a radical? radicand( the number inside the square root) has no square factors. === ** Example 1. ** 33, for example, has no square factors. Its factors are 3**·** 11, neither of which is a square number. Therefore, is simplified, or, as we say, in its simplest form. ===

 [|rational exponents], Lesson 29. Therefore, = = **·**  = 3. We have simplified. Example 3. Simplify. 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 31 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 4. Simplify. 180 = 2**·** 90 =2**·** 2**·** 45 = 2**·** 2**·** 9**·** 5 = 2**·** 2**·** 3**·** 3 **·** 5 Therefore, = 2**·** 3 = 6. Problem 1. Simplify the following. Inspect each radicand for a square factor: 4, 9, 16, 25, and so on. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). Do the problem yourself first! a) =  b)  =  =  = 5 c) =  =  = 3 d)  =  = 7 e) =  = 4 f)  =  = 10 g) =  = 5 h)  =  = 4    2 || = || 2 || = || 2 || = || ||
 * Here is a simple illustration: || [[image:http://www.themathpage.com/alg/alg_img/515.gif width="298" height="30"]] || ) ||
 * Solution**. We have to factor 42 and see if it has any square factors. We can begin the factoring in any way. For example,
 * Solution.**
 * a) || [[image:http://www.themathpage.com/alg/Alg_IMG/sq20U.gif width="29" height="21"]]

3 || = || 3 || = || 3 || = || 2 || 2 || = || The radical is in its simplest form. The fraction cannot be reduced. ||  
 * b) || [[image:http://www.themathpage.com/alg/Alg_IMG/sq72U.gif width="29" height="21"]]
 * c) || [[image:http://www.themathpage.com/alg/Alg_IMG/sq22U.gif width="29" height="21"]]

 

 
 * b) 4[[image:http://www.themathpage.com/alg/Alg_IMG/sq75.gif width="29" height="21"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq147.gif width="37" height="21"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq3.gif width="22" height="17"]] || = || 4[[image:http://www.themathpage.com/alg/Alg_IMG/249.gif width="53" height="17"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/250.gif width="53" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||
 * || = || 4**·** 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] − 2**·** 7[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||
 * || = || 20[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] − 14[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||
 * || = || 7[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||
 * || = || 20[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] − 14[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||
 * || = || 7[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||
 * || = || 7[[image:http://www.themathpage.com/alg/Alg_IMG/sq3Gr.gif width="25" height="17"]] ||


 * c) 3[[image:http://www.themathpage.com/alg/Alg_IMG/sq28.gif width="29" height="21"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq88.gif width="29" height="21"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq112.gif width="38" height="21"]] || = || 3[[image:http://www.themathpage.com/alg/Alg_IMG/251.gif width="40" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/252.gif width="53" height="17"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/253.gif width="53" height="17"]] ||
 * || = || 3**·** 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] + 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq22Gr.gif width="30" height="16"]] − 2**·** 4[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] ||
 * || = || 6[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] + 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq22Gr.gif width="30" height="16"]] − 8[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] ||
 * || = || 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq22Gr.gif width="30" height="16"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] ||
 * || = || 6[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] + 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq22Gr.gif width="30" height="16"]] − 8[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] ||
 * || = || 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq22Gr.gif width="30" height="16"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] ||
 * || = || 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq22Gr.gif width="30" height="16"]] − 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq7Gr.gif width="22" height="17"]] ||

  2 || = || 2 || = || 2 −, || on [|dividing] each term in the numerator by 2. (Lesson 20) || [|Problem 2]. 5 || = || 5 || = || 2 + ||
 * d) 3 + [[image:http://www.themathpage.com/alg/Alg_IMG/sq24.gif width="29" height="21"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq54.gif width="29" height="21"]] || = || 3 + [[image:http://www.themathpage.com/alg/Alg_IMG/254.gif width="40" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/255.gif width="40" height="17"]] ||
 * || = || 3 + 2[[image:http://www.themathpage.com/alg/Alg_IMG/sq6Gr.gif width="22" height="17"]] + 3[[image:http://www.themathpage.com/alg/Alg_IMG/sq6Gr.gif width="22" height="17"]] ||
 * || = || 3 + 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq6Gr.gif width="22" height="17"]] ||
 * || = || 3 + 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq6Gr.gif width="22" height="17"]] ||
 * || = || 3 + 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq6Gr.gif width="22" height="17"]] ||
 * e) 1 − [[image:http://www.themathpage.com/alg/Alg_IMG/sq128.gif width="38" height="21"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/sq18.gif width="29" height="17"]] || = || 1 − [[image:http://www.themathpage.com/alg/Alg_IMG/256.gif width="53" height="17"]] + [[image:http://www.themathpage.com/alg/Alg_IMG/257.gif width="40" height="17"]] ||
 * || = || 1 − 8[[image:http://www.themathpage.com/alg/Alg_IMG/sq2Gr.gif width="22" height="17"]] + 3[[image:http://www.themathpage.com/alg/Alg_IMG/sq2Gr.gif width="22" height="17"]] ||
 * || = || 1 − 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq2Gr.gif width="22" height="17"]] ||
 * || = || 1 − 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq2Gr.gif width="22" height="17"]] ||
 * || = || 1 − 5[[image:http://www.themathpage.com/alg/Alg_IMG/sq2Gr.gif width="22" height="17"]] ||
 * a) || [[image:http://www.themathpage.com/alg/Alg_IMG/258.gif width="57" height="21"]]
 * b) || [[image:http://www.themathpage.com/alg/Alg_IMG/260.gif width="70" height="20"]]

6 || = || 6 || = || 3 || on dividing [|each term] by 2. ||
 * c) || [[image:http://www.themathpage.com/alg/Alg_IMG/262.gif width="60" height="20"]]