Make the radical parts similar when they are not, subtract the coefficient of the subtrahend from that of the minuend, and prefix the difference to the common radical. If the radical parts are not and cannot be made similar, indicate the subtraction by connecting them with the proper sign.

K

CASE VI.

159. To multiply radicals.

1. Multiply 3√α by 5 No√√b.

OPERATION.

3√√a × 5No√√b= 3 × 5 × √ a × √√b = 15 √ ab

As it makes no difference in what order the factors are taken, we unite in one product the numerical coefficients; and va× √b = √ ab (Art. 143, Note 2).

=

Multiplying as in the preceding examples, we have a3, or a +; i. e. the index of the product is the sum of the indices of the factors.

but

=

From these examples we deduce the following

RULE.

I. Reduce the radical parts, if necessary, to equivalent radicals having a common index, and to the product of the radical parts placed under the common radical sign prefix the product of their coefficients.

II. Roots of the same quantity are multiplied together by adding their fractional indices.

10. Multiply 2a + b by 6 x ✔a+b.

Ans. 12x(a+b).

11. Multiply x, x, and ✔x together.

12. Multiply 3 by 28.

13. Multiply a√x1y by bNo√xy.

Ans. 1

Ans. 62.

Ans. aby.

14. Multiply (a+b)a by (a—b). Ans. (a2 — b2)3.

15. Multiply √‡ by 3 No7.

16. Multiply 25 by 4 No8.

produce the dividend, the coefficient of the quotient must be a number which, multiplied by 4, will give 60, the coefficient of the dividend, i. e. 15; and the radical part of the quotient must be a quantity which, multiplied by √ 5x, will give √ 15x, i. e. 3; the quotient required, therefore, is 15 √3.

2. Divide 64y by 22y.

OPERATION.

6√4y÷2√2y = 6 √64 y3 ÷ 24y2 = 3√ 16 y

We reduce the radical parts to equivalent radicals having a common index (Art. 155), and then divide as in the preceding example. 3. Divide a by

a.

Dividing as in the preceding examples, we have ya, or a. But } = 1 − }; i. e. the index of the quotient is the index of the dividend minus the index of the divisor.

From these examples we deduce the following

RULE.

I. Reduce the radical parts, if necessary, to equivalent radicals having a common index, and to the quotient of the radical parts placed under the common radical sign prefix the quotient of their coefficients.

II. Roots of the same quantity are divided by subtracting the fractional index of the divisor from that of the dividend.

In accordance with the definition of involution, we take the quantity three times as a factor. By Art. 159 the product is 27 √x3.

2. Find the square of 2a.

OPERATION.

(2√√ a)2 = (2 a3)2 = 4a3

In this case we have used the fractional exponent, and found the square of the given quantity by multiplying its

exponent by the index of the required power, according to Art. 126. Hence,

RULE.

I. Involve the radical as if it were rational, and placing it under its proper radical sign, prefix the required power of its coefficient.

II. A radical can be involved by multiplying its fractional exponent by the index of the required power.