61 CHAPTER 4 RADICAL EXPRESSIONS th 4.1 The n Root of a Real Number A real number a is called the nth root of a real number b if an = b . Thus, for example: 2 is a square root of 4 since 22 = 4 . −2 is also a square root of 4 since ()−22 = 4 . 2 is a cube root of 8 since 23 = 8 . Note that 8 has no other real cube roots. −2 is a cube root of −8 since ()−23 = − 8 . Note that −8 has no other real cube roots. 2 is a fourth root of 16 since 24 = 16 . −2 is also a fourth root of 16 since ()−24 = 16 . 2 is a fifth root of 32 since 25 = 32 . Note that 32 has no other real fifth roots. −2 is a fifth root of −32 since ()−25 = − 32 . Note that −32 has no other real fifth roots. Note: 1. A real number has two nth roots when n is even and 2. only one nth root when n is odd . 3. The odd nth root of a positive number is positive and 4. the odd nth root of a negative number is negative. We use the notation n b to denote the principal nth root of a real number b. The above notation is called a radical expression , b is called the radicand , n is the index of the radical, and the symbol is called the radical sign . The index n is omitted when the index is 2 . The principal root is chosen to be the positive root for the case when the index is even ; it is chosen to be the unique root when n is odd . For example, 42=3 82 =4 162 = 5 322 = 3−=−82 5 −=− 322 Note: n b is undefined in the set of real numbers when b is negative and n is even. Thus, we say that −36 is undefined over the reals or is not a real number . 62 4.2 Simplifying Radical Expressions There are three conditions for a radical expression to be in simplest form : 1. There are no perfect n power factors under a radical of index n. 2. There can be no fractions under a radical sign OR there can be no radicals in the denominator. 3. The index should be the smallest possible. In this section we will explain how to accomplish the first condition. We will use the properties nab= n a ⋅ n b for n ≥ 2 an a n = for n ≥ 2 b n b where we assume that a and b are such that the expressions do not become undefined. Let us first consider numerical radicands. Examples 1. 98 Solution: 98 is not in simplest form there is a perfect square factor 49 under the radical sign: 98=49 ⋅ 2 OR 7 2 ⋅ 2 =⋅492 =⋅ 7 2 2 =72 = 7 2 2. 3 16 Solution: 3 16 is not in simplest form since there is a perfect cube factor 8 under the radical sign: 316= 3 8 ⋅ 2 OR3 2 3 ⋅ 2 =⋅3 832 =⋅3 2 3 3 2 =232 = 2 ⋅ 3 2 3. 4 80 Solution: 4 80 is not in simplest form since there is a perfect 4 th power factor 16 under the radical sign: 480= 4 16 ⋅ 5 OR4 2 4 ⋅ 5 =⋅416 4 5 =⋅4 2 4 4 5 =24 5 = 2 4 5 Next, let us consider variable radicands . Let us also first consider the case when the index n is odd . We observed in Section 4.1 that odd nth roots follow the sign of the radicand, thus 63 n xxnn = for odd and any real number x . For example, 3 xx3=, 5 yy 5 =, 7 mm 7 = , and so on. Next, let us consider the case when the index n is even . We defined in Section 4.1 the principal nth root to be the positive root when n is even. Consider the following: 22 = 2 and also () −22 = 2 4 24 = 2 and also4 () − 24 = 2 From the above we conclude that for any real number x and even index n, n xn = x if x is positive, and n xn = − xif x is negative which can be abbreviated to n xn = x for any real number x . Thus, for example: x2 = x 4 w4 = w 6 c6 = c If we want to remove the cumbersome absolute value notation we need to make the assumption that all variables represent positive real numbers . Thus, with this assumption we will have x2 = x 4 w4 = w 6 c6 = c Let’s now simplify radicals with variable radicands. We will assume that the variables can be any real number . Examples 4. Simplify: x 4 Solution: x4= x 2 = x 2 6 5. Simplify: y 6 3 Solution: y= y 64 12 6. Simplify: a 12 6 6 Solution: a= a = a 7. Simplify: n22 Solution: n22= n 11 8. Simplify: 3 x 4 Solution: 3x5 = 3x3 ⋅= x 3 x 3 ⋅3 xx = x 3 9. Simplify: 3 m23 Solution: 33m23=m21 ⋅= m 2 33 m 21 ⋅ m 2 = m 7 3 m 2 10. Simplify: 4 p8 Solution: 4 p8= p 2 = p 2 11. Simplify: 4 w 36 Solution: 4 w36= w 9 12. Simplify: 5 n12 Solution: 55n12=n10 ⋅= n 2 55 n 10 ⋅ nn 2 = n 2 5 2 13. Simplify: 6 x18 Solution: 6 x18= x 3 Now let us consider a combination of numbers and variable factors for radicands. We will continue to assume that the variables can represent any real number. 14. Simplify: 32 x4 y 6 Solution: 32x4 y 6 =16xy46 ⋅= 2 16xy 46 ⋅= 2 4xy 23 242 = x2 y 3 15. Simplify: 128 a10 b 16 c 2 Solution: 128a10 b 16 c 2 =64abc10162 ⋅= 2 64abc 10162 ⋅= 2 8abc 58 282 = a5 c b 8 16. Simplify: 3 54 n5 p 10 Solution: 3354np510=27np39 ⋅= 2 np 2 33 27np 39 ⋅ 2 np 2 = 3np 3 3 2 np 2 65 17. Simplify: 3 375 x32 y 46 z 21 Solution: 3 375xyz32 46 21=3 125xyz304521 ⋅= 3 xy 2 3 125xyz 304521 ⋅=3 3 xy2 5xyz 10157 3 3 xy 2 18. Simplify: 4 162 a8 m 12 e 26 Solution: 4162ame81226= 481ame81224 ⋅= 2 e 2 44 81ame 81224 ⋅ 2 e 2 =3 a2 m 3 e 6 42e2 = 3 amm 263 4 2 e 2 19. Simplify: 4 16 y30 n 34 Solution: 4 16yn3034=416yn2832 ⋅= 2 yn 22 4 16yn 2832 ⋅=4 2 yn 22 2yn 78 4 222 ynynyn 22 = 7822 4 Next, we have examples showing what to do when there are factors outside the radical expression. 20. Simplify: 3ab25 64 a 6 b 13 Solution: 3643ab261325 ab= ab 5 32a5 b 10 ⋅= 23 ab 32 ab5 32a 5 b 10 ⋅ 5 2 ab 3 =3ab2 ⋅2ab2 ⋅5 262 ab 324 = a b 5 ab 3 21. Simplify: −4w26 64 y 18 w 36 Solution: −4wyww218366 64 =− 4 26 64y18 w 36 =−⋅ 4 w 2 2 y 3 w 6 =− 8 wy 83 Finally, let us look at examples involving rational radical expressions. For these we will assume that the variables represent nonzero real numbers. 63 a16 b 8 22. Simplify: 25 c22 63ab16 8 9a168 b⋅ 7 3 a 84 b 3 7 ab8 4 Solution: = =7 = 25c 22 25c22 5 c 11 5 c11 40 m7 23. Simplify: 3 27 n15 40m7 8m6⋅ 5 m 2m 2 Solution: 3= 3 = 3 5m 27 n15 27n15 3n 5 2 48 x18 y 42 24. Simplify: 4 xy4 z z 4 Solution: 248xy18 42 216xy16 40⋅ 32 xy 2 2 2xy4 10 4 xy3 6 4= 4 =⋅43xy22 = 4 3 xy 22 xyzz4 4 xyz 4 z4 xyz4 z zz 66 4.3 Adding or Subtracting Radical Expressions We can add or subtract only like radicals , i.e., those radical expressions that have the same index and the same radicand . Examples Perform the operations. 1. 23+3 3 + 73 − 43 3 3 3 33 3 Solution: 23++ 373432373 − = + +− 34393 =− 33 2. xy− yx4 −4 xy + 4 yx 4 Solution: xyyx−−444 xy + yxxy 4 =− 4 xyyx −+ 44 433 yx =− xy + yx 4 Recall that sometimes the radicals need to be simplified first. In the following assume that the variables represent positive real numbers . 3. 8y3 − 4 y 18 y Solution: 8418yyy3 − =4y2 ⋅− 24 yyy 9 ⋅= 2 2y 24 yyy −⋅ 3 2 =2yy 2 − 12 yy 2 =− 10 yy 2 4. 6483a5+ a 3 6 a 2 Solution: 648333aaa52+ 66 =8a3 ⋅+ 6 aaa 22 3 66 =⋅ 2a 33 6 aaa 22 + 6 =12aaaa3 62 + 3 6 2 = 136 aa 3 2 5. 3484 xy5 − 6 x4 243 xy Solution: 3484 x5 y− 6243 x4 xy = 34 16x4 ⋅−⋅ 3 xy 6 x4 81 3 xy =⋅32x4 3xy −⋅ 6 x 3 44 3 xy = 63 x xy − 183 x 4 xy =− 123 x 4 xy 6. 5−32ab711 − 4 ab 5 ab 26 5 Solution: 5−324ab711 − abab 5 26 =−5 ()2ab5 10 ⋅− ab 2 4 abab5 2 b 5 5 =−5 ()2ab510 ⋅−5ab24 ab 55 b 5 ⋅=− ab 2 24 ab 22 5 ab − ab 22 5 ab =− 6 ab 22 5 ab 67 4.4 Multiplying Radical Expressions to multiply radicals with the same index n ≥ 2 we use the following: na⋅ n b = n ab . Examples 1. Multiply: 12⋅ 30 Solution: 12⋅ 30 =⋅= 1230 360 =36 ⋅= 10 6 10 Sometimes when the numbers are large it is better to use prime factorization rather than carry out the multiplication.