and as the quantity under the radical sign of the second degree is a perfect square, its root can be extracted: hence,

that is, when the index of a radical is a multiple of any number n, and the quantity under the radical sign is an exact n' power, We can, without changing the value of the radical, divide its inder by n, and extract the nth root of the quantity under the sign.

153. Conversely, The index of a radical may be multiplied by any number, provided we raise the quantity under the sign to a power of which this number is the exponent.

For, since a is the same thing as " a", we have,

154. The last principles enable us to reduce two or more radicals of different degrees, to equivalent radicals having a common index.

For example, let it be required to reduce the two radicals 3/2a and (a + b)

to the same index.

4

By multiplying the index of the first by 4, the index of the second, and raising the quantity 2a to the fourth power; then multiplying the index of the second by 3, the index of the first, and cubing a+b, the value of neither radical will be changed, and the expressions will become.

3/2a = 12/21a2 = 12/16a; and √(a+b) = 12 / (a + b)3, and similarly for other radicals: hence, to reduce radicals to a common index, we have the following

RULE.

Multiply the index of each radical by the product of the indices of all the other radicals, and raise the quantity under each radical sign to a power denoted by this product.

This rule, which is analogous to that given for the reduction of fractions to a common denominator, is susceptible of similar modifications.

For example, reduce the radicals

Since 24 is the least common multiple of the indices, 4, 6, and 8, it is only necessary to multiply the first by 6, the second by 4, and the third by 3, and to raise the quantities under each rad ical sign to the 6th, 4th, and 3d powers, respectively, which gives √ a=24/a; °/5b=24 / 51b*, 3/a2 + b2 = 24 / (a2 + b2)3. √56

8

Addition and Subtraction of Radicals of any Degree.

155. We first reduce the radicals to their simplest form by the aid of the preceding rules, and then if they are similar, in order to add them together, we add their co-efficients, and after this sum write the common radical; if they are not similar, the addition can only be indicated.

1. Find the sum of√48ab2 and b√75a. 2. Find the sum of 3/4a2 and 23/2a.

Ans. 9b3a.

Ans. 53/2a.

3. Find the sum of 2√45 and 35:

Ans. 95.

155*. In order to subtract one radical from another when they are similar,

Subtract the co-efficient of the subtrahend from the co-efficient of the minuend, and write this difference before the common radical

Thus,

but,

2abcd -5ab√c are irreducible.

1. From 3/8a3b+16a subtract 3/b1 + Qab3.

Ans. (2a - b) 3/b+ 2a.

2. From 3/4a2 subtract 23/2a.

Ans. 3/2a.

Multiplication of Radicals of any Degree

156. We have shown that all radicals may be reduced to equivalent ones having a common index; we therefore suppose this transformation made.

Now, let ab and cd denote any two radicals of the same degree. Their product may be denoted thus,

or since the order of the factors may be changed without affecting the value of the product, we may write it,

ac × "√b × "√d x n bxd

we have finally,

or (Art. 150), since "x"="/db;

hence, for the multiplication of radicals of any degree, we have the following

RULE.

I. Reduce the radicals to equivalent ones having a common index. II. Multiply the co-efficients together for a new co-efficient; after this write the radical sign with the common index, placing under it the product of the quantities under the radical signs in the two factors; the result is the product required.

2. The product

3a 4/8a2 × 2b1/4u3c = 6ab 1/ 32a1c = 12a2b 4/2c.

3

5. Multiply √2×2√3 by √ √

6. Multiply 2/15 by 33/10.

Ans. 6/337500.

Ans. 8

8. Multiply √2, 3/3, and 4/5, together.

Division of Radicals of any Degree.

157. We will suppose, as in the last article, that the radicals have been reduced to equivalent ones having a common index. Let ab and cd represent any two radicals of the n' degree. The quotient of the first by the second may be written,

Hence, to divide one radical by another, we have the fol lowing

RULE.

I. Reduce the radicals to equivalent ones having a common index.

II. Divide the co-efficient of the dividend by that of the divisor for a new co-efficient; after this write the radical sign with the common index, and place under it the quotient obtained by dividing the quantity under the radical sign in the dividend by that in the divisor; the result will be the quotient required.

EXAMPLES.

1. What is the quotient of e3/a2b+b divided by d

c

2. Divide 2/3 x 3/4 by 2×3/3.