50. Quantities having different indices may be reduced to equivalent quantities that shall have the same index, by reducing the indices to a common denominator, then raising each of the quantities to the power denoted by the numerator of its index, and indicating the root denoted by the common denominator. Thus, a + b = a + b3=(a3)' + (b2)s. 2}+3=2}+33= (23)' + (3o)' = 8' + 93.

:

EXERCISES.

Reduce the following quantities to others having a common index:

ADDITION AND SUBTRACTION OF SURDS.

51. To Add or Subtract Surds; Reduce them if necessary to their simplest form, and proceed as with rational quantities.

(6.) 32a x-108.x1. √32a3x

Ans. (2a+3x)V4x.

(7.) (a3+2a2x+ax2)1+(a3—2a2x+ax2)}. 2a1.

(8.) 1/243x-768x+√48x.

4/3x.

MULTIPLICATION AND DIVISION OF SURDS.

52. The multiplication and division of surds are conducted on the same principles as the multiplication and division of rational quantities. Thus Jax ba11=(ab)1; 21×3 =61; x1×x3=x3; 21× 3 = 81× 9=(72)'.

(1.) √2 × √5.

(2.) 2/3×35.

(10.) {(a + x)1 +(a—x) } } × { (a + x)'—(a-x)}}. 2x.

(11.) (10+√19) × (10 — √√19)3.

(12.) (Na + √b + √c)× (− ja+√b+√c)×(√a −√o+√e) × (√a+ √b−√e).

3.

-a-b2 — c2+2(ab+ ac+be).

Surds to be divided:

(13.) √20√5.

(14.) 43/8132/3.

Ans. 2.

D

4.

(18.) (x+1+1)÷ (x3 −1+11). . x2+1+x−1. (19.) {(1—2)2 +(1+x)}+{1+___;}. (1−x)*. (20.) (Vā3—N ̄3)÷ (Na—1/6).

a1+(ab)1+b1.

53. The square root of a quantity cannot be partly rational and partly irrational. For if possible let x = a + √b, then by squaring these equals, x=a2+2a√b+b, and hence which is impossible.

=

x— a 2-b2

2a

54. From this it follows that if an equation be of the form x+y=a+b+c, the rational parts must be equal, and also the irrational parts, viz, x = a + b, and Y = √c.

55. If a binomial surd be of the form a+ √b, one term being rational, the square root may be extracted thus: Assume (a+b) = √x+√Y

then by squaring, a+√b=x+y+2√xy
and hence, (54.) x+y=a, and 2√xy = √b.
Squaring each of these equals,

we get x+2xy+y=a2; and 4xyb

* .. x2-2xy +y2a2-b; and x-ya-b.

·· √x+√y or (a+ √b)3 = (a+Va3—b)},

2

The sign.. is sometimes used as an abbreviation for "There

fore."

It thus appears that when a2 b is a perfect square, the square root of a+b may be expressed by a binomial, which may be found by assuming it = √x+ √y, and proceeding as above, or by substitution in the general formula.

If the surd be of the form a

-y, and proceed as before.

-

b, assume the root = √x

If the surd be of the form a c√bc, or can be put under the form c(a+b), its square root may be found by first finding the root of a√b, and then multiplying the result by /c.

EXERCISES.

56. A fractional surd may in most cases be much simplified by multiplying numerator and denominator by such a quantity as will render the denominator rational. Thus, if

the denominator be of the form bn, the multiplier is b1- n; if it be of the form ab, the multiplier is ab1⁄2 ; if it be of the form a3±3, the multiplier is aa‡ab+b2; and if it be of the form a±b, first multiply by aa = b2, then the result ab multiplied by a+b will be rational.

A general formula for all such multipliers may easily be proved, but it is seldom of use.

EXERCISES.

Express each of the following fractions with a rational.

EQUATIONS CONTAINING SURDS.

57. When an equation contains a surd, if, by transposition or other operations, the surd be made to stand alone on one side of the equation, and both members be then raised to the power corresponding to the root expressed in the surd, an equation will be obtained which will be free of surds.

If the equation contain more than one surd, the same process must be repeated as often as may be necessary.