Ex. 1. Find the square root of

4x-12 + 29 x2 - 30 x + 25.

The work follows:

4x4-12 x3 + 29 x2 - 30 x + 25 | 2 x2 − 3 x + 5

20 x2-30 x + 25 4x2-6x+5

Only the trial divisor and the complete divisor of each stage are written, the other steps being performed mentally.

4. The preceding method can be extended to find square roots which are multinomials of any number of terms.

The work consists of repetitions of the following steps:

After one or more terms of the root have been found, obtain each succeeding term, by dividing the first term of the remainder at that stage by twice the first term of the root.

Find the next remainder by subtracting from the last remainder the expression (2 a + b) b, wherein a stands for the part of the root already found, and b for the term last found.

EXERCISES III.

Find the square root of each of the following expressions:

17. x6 - 6 ax5 + 15 a2x2 - 20 a3x3 + 15 a2x2 6 αχ + α.

32 a3 +48 a4 + 12 a2.

19. 4 a6 + 17 a2 — 22 a3 + 13 a4 — 24 a − 4 a5 + 16.

20. 9 x6 + 6 x3y + 43 x1y2 + 2 x3y3 + 45 x2y1 — 28 xy5 + 4 yo.

22. a2mx2n + 10 a2m-2x2n+1 6 am+1xn+1 + 25 ɑ2m-4x2n+2 - 30 am-12n+2

+ 9 a2x2.

§ 4. CUBE ROOTS OF MULTINOMIALS.

1. The process of finding the cube root of a multinomial is the inverse of the process of cubing the multinomial.

(i.) The first term of the root is the cube root of the first term of the multinomial; i.e., a = V/a3.

(ii.) If the cube of the first term of the root be subtracted from the multinomial, the remainder will be

3a2b+3ab2+ b3, (3a2 + 3 ab+b2)b.

= =

Three times the square of the first term of the root, 3 a2, is called the Trial Divisor.

(iii.) The second term of the root is obtained by dividing the

first term of the remainder by the trial divisor; i.e., b

3a2b

3 a2

(iv.) If the sum 3 a2+3 ab + b2, the complete divisor, be multiplied by the second term of the root, and this product be subtracted from the first remainder, the second remainder will be 0.

(4)

3a2b+3ab2+b3 | = (3a2+3 ab+b2) ×b

Ex. 1. Find the cube root of 27 x3 + 54 x2y + 36 xy2 + 8 y3.
The work, arranged as above, is:

27 x2+54 x2y+36 xy2+8 y3 | 3x+2y

27 203

54 x2y

3(3x)2=27 x2, trial divisor
54x2y÷27x2=2y, second term of

(1)

3 (3 x)2+3 (3 x) (2 y) +(2 y)2=27 ∞2
+18 xy+4y2, complete divisor (3)

54x2y+36xy2+8 y3 | = (27 x2+18 xy +4 y2) (2 y)

(4)

2. The preceding method can be extended to find cube roots which are multinomials of any number of terms, as the method of finding square roots was extended. The work consists of repetitions of the following steps:

After one or more terms of the root have been found, obtain each succeeding term by dividing the first term of the remainder at that stage by three times the square of the first term of the root. Find the next remainder by subtracting from the last remainder the expression (3 a2 + 3 ab + b2)b, wherein a stands for the part of the root already found, and b for the term last found.

The given multinomial should be arranged to powers of a letter of arrangement.

Ex.

27-27x+90 x2-55 x3 +90 x4-27 x5+27 x6 | 3−x+3x2

27

-27 x

-27x+9x2 — 23

81 x2-54x3

3(3)2=27

3(3)2+3(3)(−x) + ( − x)2 = 27—9x+x2 3(3-x)2+3(3−x) (3 x2) + (3 x2)2=

81 x2-54x3+ 90 x4-27x5 +27 x6 27-18x+30 x2-9x3+9x4

EXERCISES IV.

Find the cube root of each of the following expressions:

1. x6 6x5 + 15 x1 — 20 x2 + 15 x2 - 6 x + 1.

8. 8a6+ 48 a5b + 60 a1b2 — 80 a3b3 — 90 a2b4 + 108 ab5 — 27 bo.

9. x3-3x2 - 9x11 27 x15 – 6 x5 - 54 x13 + 28 x9.

10. 108 a5 - 48 aa + 8 a3 + 54 a7 — 12 a8 + a9 112 a6.

108 ax5 - 90 a2x2 + 80 a3x3.

12. 13x

— 3 x8 — x9.

14. 64x3n-144 x3n−1+12 x3n−2+117 x3n−3–6 x3n—4—36 x3n−5 – 8 x3n−6 ̧

1. Since

§ 5. HIGHER ROOTS.

N√√N, wherein N stands for any multinomial, the fourth root is most easily found as the square root of the square root of the given multinomial.

In like manner, since N=√N, the sixth root can be found as the cube root of the square root of the given multinomial. And so on for any root whose index can be factored.

2. The process of finding the nth root of a multinomial is the inverse of raising a multinomial to the nth power.

The method can be derived from the expression for (a + b)", which will be given in Ch. XXVIII.

EXERCISES V.

Find the fourth root of each of the following expressions:

1. x8+ 4x6 + 6 x2 + 4x2 + 1.

2. a8 +4 ab + 10 a6b2 + 16 a5b3 + 19 a1b1 + 16 a3b5 + 10 a2bε + 4 ab2 + 68.

3. 16x8. 160 x 408x6 +440x5 - 2111 x4 - 1320 x3 + 3672 x2 + 4320 x

+ 1296.

4. 625x8 +5500 x + 17150 x6 + 20020 x5 + 721 x1 . 8008 x8+2744 x2

- 352 x + 16.

Find the sixth roots of each of the following expressions :

5. 64x12-192x10 + 240 x8 - 160 x6 + 60x4 12 x2 + 1.

6. a126 a11b +21 a1ob2 + 50 aob3 + 90 a8b4 + 126 a7b5 + 141 ab® + 126 a5b7+ 90 a4b8 + 50 a3b9 + 21 a2b10 +6 ab11 + b12.

§ 6. ROOTS OF ARITHMETICAL NUMBERS.
Square Roots.

1. Since the squares of the numbers 1, 2, 3, 9, 10, are 1, 4, 9, ..., 81, 100, respectively, the square root of an integer of one or two digits is a number of one digit.

Since the squares of the numbers 10, 11, ..., 100, are 100, 121, ..., 10000, the square root of an integer of three or four. digits is a number of two digits.

In general, the square root of any integer of 2 n − 1 or 2 n digits is a number of n digits.

Therefore, to find the number of digits in the square root of a given integer, we first mark off the digits from right to left in groups of two. The number of digits in the square root will be equal to the number of groups, counting any one digit remaining on the left as a group.

2. The method of finding square roots of numbers is then derived from the identity

(a + b)2= a2 + (2 a + b) b,

(1)

wherein a denotes tens, and b denotes units, if the square root be a number of two digits.

Ex. 1. Find the square root of 1296.

We see that the root is a number of two digits, since the given number divides into two groups. The digit in the tens' place is 3, the square root of 9, the square next less than 12. Therefore, in the identity (1), a denotes 3 tens, or 30.