10. Find the sum of 120am-20acm+5mc and 20mc 60acm+45a2m.

Ans. (c-a)V5m.

11. Find the sum of 3 cx3, ax3, and 21 ax3.

Ans. 3x(c+&a).

12. Find the sum of 5a(cx*—dx')3 and 2x(a'd—a°c)}.

14. Find the sum of √(1+a)-', Va'(1+a)~', and

252. When the radicals are similar, it is evident that we may make the common radical the unit of subtraction.

following

RULE. I. Reduce each radical to its simplest form.

Hence the

II. If the resulting radicals are similar, find the difference of the coefficients, and to the result annex the common radical part; if dissimilar, indicate the subtraction by the proper sign.

6. From (a2c3—3c3x)* take 2(a3d'—3dx)3.

Ans. (c—2d) (aa—3x)3.

7. From (a'—al2+a2b—b3)1⁄2 take (a'—3a2b+3ab2—b′)3⁄4.

253.

It has already been shown (227) that the nth root of the product of two or more factors is equal to the product of the nth roots of those factors. And since the converse of this proposition is true, we shall have

If the radicals have coefficients, the product of the coefficients may be taken separately. Thus,

cvadb = c×d×a×Ŵ/b = cdVab.

If the radicals have not a common index, they must first be reduced to the same degree.

Let it be required to find the product of ax and by.

RULE. I. If necessary, reduce the given radicals to a common index.

II. Multiply the quantities in the radical parts together, and place he product under the common radical sign; to this result prefix the product of the given coefficients, and reduce the whole to its simplest form.

254. Since a fraction is raised to any power by involving its numerator and denominator separately to the required power, it is evident that any root of a fraction will be obtained by extracting the required root of each term separately. Hence we have

The quotient of the nth roots of two quantities is equal to the nth root of their quotient.

Upon this principle is based the rule for the division of radicals.

RULE. I. If necessary, reduce the radicals to a common index.

II. Divide the coefficient of the dividend by the coefficient of the divisor; divide also the quantity in the radical part of the dividend by the quantity in the radical part of the divisor, placing the quotient under the common radical sign. Prefix the former quotient to the latter, and reduce the result to its simplest form.

POWERS AND ROOTS OF RADICAL QUANTITIES.

255. According to the rule for multiplication of radicals, to

form the mth power of a",

ora, we must take the quantity, a,

m times as a factor, and affect the result by the common radical index. Hence,

The mth power of the nth root of a quantity is equal to the nth root of the mth power of that quantity.

1

256. To obtain the mth root of the radical a" may proceed as follows: Let

Involving both members of (1) to the mth power,

1

x = or Va;

involving both members of (2) to the nth power,

(2)

(8)

(4)

The mth root of the nth root of a quantity is equal to the math root of that quantity.

The mth root of the nth root of a quantity is the same as the nth root of the mth root of that quantity.