Transformation of Radicals of any Degree.

150. The principles demonstrated in Art. 104, are general. For, let a and b, be any two radicals of the nth degree,

and denote their product by p. We shall have,

By raising both members of this equation to the n power, we find

whence, by extracting the n root of both members,

Since the second members of equations (1) and (2) are the same, their first members are equal, whence,

1st. The product of the n' roots of two quantities, is equal to the nth root of the product of the quantities.

Denote the quotient of the given radicals by q, we shall have

and by raising both members to the nt power,

(1);

whence, by extracting the n' root of the two members, we have,

The second members of equations (1) and (2) being the same, their first members are equal, giving

2d. The quotient of the nth roots of two quantities, is equal to

the nth root of the quotient of the quantities.

151. Let us apply the first principle of article 150, to the simplification of the radicals in the following

EXAMPLES.

1. Take the radical 3/54a4b3c2. This may be written,

3 54a463c2 = 3/27a3b3 × 3/2ac2 = 3ab 3/2ac2.

6 192a7bc12

6

6/64a6c12 X 3ab = 2ac26/3ab.

In the expressions, 3ab3/2ac2, 23a2, Qab2c 4/3uc2, each quantity placed before the radical, is called a co-efficient of the radical.

Since we may simplify any radical in a similar manner, we have, for the simplification of a radical of the nth degree, the following

RULE.

Resolve the quantity under the radical sign into two factors, onc of which shall be the greatest perfect nth power which enters it; extract the nth root of this factor, and write the root without the radical sign, under which, leave the other factor.

Conversely, a co-efficient may be introduced under the radicul sign, by simply raising it to the nth power, and writing it as a factor under the radical sign.

Thus, 3ab 3/2ac2 = 3/27a3b3 × 3/2ac2 = 3/54a1b3c2.

152. By the aid of the principles demonstrated in article 143, we are enabled to make another kind of simplification.. Take, for example, the radical /4a2; from the principles re ferred to, we have,

6

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

and the quantity under the radical sign is an exact n2 power, We can, without changing the value of the radical, divide its index 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

we have,

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

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

to the same index.

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

=

4

12

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 °/5b = 24 / 51b4, 3/ a2 + b2 = 21 / (a2 + b2)3.

4

* √ a = 24/a;

6

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.

Thus,

33/6+23/6=53/6.

EXAMPLES.

1. Find the sum of 48ab2 and b75a.

Ans. 9b3a.

2. Find the sum of 36/4a2 and 23/2a.

Ans. 53/2a.

3. Find the sum of 245 and 3√5.

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

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 a and c 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,

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.