2 FIG. 77. 9. Figure 77 represents a triangle each of whose three sides is 2 inches long. What is the area? [HINT. Here it is necessary to find the altitude h. Note that it is a side of a right triangle whose hypotenuse is 2 inches and whose other side is 1 inch.] 10. Find the ratio between the radius of a sphere and the radius of a second sphere whose volume is 7 times that of the first. [HINT. The volumes of any spheres are to each other as the cubes of their radii. See § 115.] 142. Simplification of Radicals. We know that the square root of the product of two numbers is the same as the product of their square roots. For example, √4X25 is the same as √4X √25 because both are equal to 10. (Explain.) In the same way, 8×3 may be written V8XV3, or simply 23. In fact, we have the following general formula : Again, is the same as because both are equal to 3. √9 (Explain.) Similarly, V may be written Formulas X and XI enable us to simplify many radical expressions, as is illustrated by the following examples : SOLUTION. Ans. √20 a=√4 a. 5=√4 a. √5-2 a3√5. Ans. 3 √ 72 x2y* _ † (8 y°) (9 x2) _ V8 y° • V9 x2 _ 2 y2 V9 x2. Ans. 26 143. Note. It should be observed that in each of the examples above the process of simplification consists in removing from under the radical sign the largest factor of the radicand that is a perfect square, perfect cube, etc. Thus, in Example 1, the radicand, 63, is first broken up into factors in such a way that 9 (which is a perfect square) appears clearly. Similarly, in Example 2 (where we are dealing with a cube root) we first write the radicand, 32, in a form which brings out to the eye its factor 8, which is a perfect cube. The first step in all such examples is, therefore, to get the radicand broken up into factors. This requires good judgment, but becomes very easy after slight practice and experience. The pupil is especially warned that one cannot write √a+b=√a+√b. Thus, when a=4, b=9 this would give √13=2+3, which is clearly false. EXERCISES 1. By Formula X we may write √20= √4.5=2√5. By looking up V20 and V5 in the tables, show that √20 is the same as 2√5. 2. Show by the tables (as in Ex. 1) that √54=3√6. Simplify each of the following expressions. (See § 143.) 16. 2√3. Put each of the following in a form without a number written outside the radical sign: SOLUTION. 2√3=√4 · √3=√12. Ans. (Formula X.) 144. Similar Radicals. Addition and Subtraction of Radicals. Whenever two radicals with the same index have the same radicand, or can be given the same radicand by simplification, they are called similar radicals. Thus, 2√2 and 3√2 are similar radicals; so also are √2 and √32, since the last of these may be simplified into 4√2. Likewise, √3a2x and √3 b2x are similar, being equal respectively to a√3 x and b√3 x. Whenever similar radicals are added or subtracted the result may be expressed in a single term. Thus, =(12-4)√2=8√2. Ans. Likewise, 2√4 a2b+√9 a2b-V16 ab 2. 2 a√b+3 a√b-4 a√б = (4 a+3 a−4 a)√b EXERCISES Combine the radicals in the following exercises whenever possible. Check your answer in Exs. 1-4 by use of the tables. 1. √8+ √18+ √32. 2. √12+√27— √75. 3. V2+V16– 154. 4. V128-√18+ √72. 9. √32a-√8a+ √18 a. 5. √ √ √ 6 √8+ √12+√16. 7. √32 a2 - √8 a2+√18 a2. 8. V16 a3b3+ √54a3b3. 10. √2(x-y)2+ √8(x−y)2+ √18(x-y)2. 11. √2(x-y)+√8(x−y)+ √18(x − y). 12. √ +√121+√}+√π}. 13. 2√3−√12+3√27. 14. Vğ+ 145. Definition of Surds. A square root which cannot be extracted exactly is commonly called a quadratic surd, or more briefly, a surd. A surd is thus merely a radical whose index is 2 (see § 139) whose value cannot be expressed exactly. For example, √3, V, V6.5, √a+b, √x2-y2 are surds, but √9, √25, √x2+2xy+y2 are not. 146. Multiplication of Surds. To multiply one surd, √a, by another, √б, we have the principle √a. √b=√ab. This principle is, in fact, what Formula X gives when n=2. EXAMPLE 1. Find the product of √2 and 18. SOLUTION. √2. √18=√2 · 18=√36=6. Ans. EXAMPLE 2. Multiply √3+√5 by 2√3−√5. EXAMPLE 3. Multiply Va+Va-b by Va-Va-b. Simplify each of the following expressions as far as possible. 1. √3.√27. 2. √8.√12. 7. V.1.V.001 9. (2√3-√2)(2√3+√2). 8. (√3−√2)(√3+√2). 10. (√5–√2)2. 11. (√5-2)2. 12. (2√5+3√3) (4√5-5√3). 13. (√2+√3+√5)(√2− √3). 14. √x.√x3. 15. √ab.√a3b3. 16. √x2у•√xy. 17. (√x+√y)(√x−√y). 18. (√a+√b)2. 19. (√3x+ V4 y)2. 20. (3√x+4√y) (2√x−5√y). 21. Find the value of x2 if x = √3+ √2. 22. Find the value of x2-4x-1 if x=2+√5. 23. Find the value of x2+3x-2 if x=(√17-3)/2. 24. Does √3-V2 "satisfy" the equation x2--4x+1=0; that is, is this equation true when x= √3+ √2? |