EXAMPLES.

1. Required the square root of a 2a3x + 3a2x2

-

a* - 2a3x+3a2x2 − 2ax2 + x*(a2 ¬ ax + x2

a* - 2a3x+3a2x2 - 2αx3 + X*

2. Required the cube root of x+6x-40x + 96.x-64.

x+6x-40x96x-64(x2 + 2x-4

3. Required the square root of a2+2ab+2ac + b2 + 2bc + c2.

4. Required the cube root of x-5x+15x* -20x3 + 15x2-6x + 1.

5. Required the 4th root of 16a-96a3x+ 216a2x2-216ax2 + 81x*.

6, Required the 5th root of 32x3 — 80x* + 80x3 — 40x2+10x-1.

OF IRRATIONAL QUANTITIES, OR SURDS.

(1) IRRATIONAL quantities, or surds, are such as have no exact root, being usually expressed by means of the radical sign, or by fractional indices; in which latter case, the numerator shows the power the quantity is to be raised to, and the denominator its root. Thus, 2, or 2*, denotes the square root of 2; and a, or a, is the square of the cube root of a, &c. (y)

CASE I.

To reduce a rational quantity to the form of a surd.

RULE.

Raise the quantity to a power corresponding with that denoted by the index of the surd; and over this new quantity place the radical sign, or proper index, and it will be of the form required.

EXAMPLES.

1. Let 3 be reduced to the form of the root.

Here 3 x 332=9; whence 9 Ans.

square

(y) A quantity of this kind, as for instance 2, is called an irrational number, or a surd, because no number, either whole or fractional, can be found, which, when multiplied by itself, will produce 2. But its approximate value may be determined to any degree of exactness, by the common rule for extracting the square root, being and certain decimals, which never terminate.

2. Let 2 be reduced to the form of the cube

root.

··Here (2xo)3 8.xo; whence V/8a, or (8)

=

Ans.

3. Let 5 be reduced to the form of the square root.

4. Let -3x be reduced to the form of the cube root.

5. Let -2a be reduced to the form of the fourth root.

6. Let a2 be reduced to the form of the fifth root. 7. Let

root.

a

by be reduced to the form of the square

a

Note. In like manner, any rational quantity may be reduced to the form of the surd to which it is joined, and their product be placed under the same index, or radical sign.

EXAMPLES.

1. 2√2=√4 × √2=√8; and 2 1/2 = 1/8 × √/2= 16.

2. 3√a=√9× √a=√9a; and÷V4α=V÷× √4@⇒

4.a

8

4. Let

3. Let 5/6 be reduced to a simple radical form, 5a be reduced to a simple radical form, 9 be reduced to a simple radical form,

2a

5. Let 24

CASE II,

To reduce quantities of different indices, to others that shall have a given index,

RULE.

Divide the indices of the proposed quantities by

the given index, and the quotients will be the new indices for those quantities.

Then, over the said quantities, with their new indices, place the given index, and they will be the equivalent quantities required.

EXAMPLES.

1. Reduce 3 and 2 to quantities that shall have the index/

Whence (3) and (22)*, or 27* and 4*,

quantities required.

are the

2. Reduce 2 and 4 to quantities that shall have the common index

1

8

3. Reduce a2 and a to quantities that shall have the common index

4. Reduce a and b to quantities that shall have the common index

1

8

Note. Surds may also be brought to a common index, by reducing the indices of the quantities to a common denominator, and then involving each of them to the power denoted by its numerator.

EXAMPLES.

1. Reduce 3 and 4 to quantities having a common index.

2. Reduce 4 and 5 to quantities that shall have a common index,

3. Reduce a and a to quantities that shall have a common index,

4. Reduce a and b to quantities that shall have a common index.

5. Reduce a and b to quantities that shall have a common index.

CASE III.

To reduce surds to their most simple forms.

RULE.

Resolve the given number, or quantity, into two factors, one of which shall be the greatest power contained in it, and set the root of this power before the remaining part, with the proper radical sign between them (≈).

EXAMPLES.

1. Let 48 be reduced to its most simple form.

2. Let 108 be reduced to its most simple form. 2/108=27 × 4=3/4 Ans.

(2) When the given surd contains no factor that is an exact power, it is already in its most simple form. Thus, 15 cannot be reduced lower, because neither of its factors, 5, or 3, is a square.