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2. Required the cube root of x+6x5-40x3+96x64.

xo+6x3 —40x3 +96x-64(x2+2x-4

3x4)6x5

x+6x5+12x4+8x3

3x1) — 12x4

x+6x5-40x396x-64

*

3. Required the square root of 4a2-12ax+9x2.

4. Required the square root of a2+2ab+2ac+b2+ 2bc+ca.

5. Required the cube root of x2-6x5+15x4-20x3+ 15x2-6x+1.

6. Required the 4th root of 16a4-96α3x+216a2x2 216αx3+81x4.

7. Required the 5th root of 32x5-80x480x3- 40x2 +10x-1.

OF IRRATIONAL QUANTITIES,
OR SURDS.

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 23, denotes the square root of 2; and Va2, or a3, is the square of the cube root of a, &c. (q)

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 square root. Here 3X3=32=9; whence 9 Ans.

(2. Reduce 2x2 to the form of the cube root.

Here (2x2)3=8x; whence 3/8x, or (8x) 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, and

va α ✔at√b, and to the form of the square root. ba

2a

Note. Any rational quantity may be reduced by the above rule, to the form of the surd to which it is joined, and their product be then placed under the same index, or radical sign.

2,

(9) A quantity of the kind here mentioned, as for instance is called an irrational number, or a surd, because no number, either whole or fractional, can be found, which when multipled 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 1 and certain non periodic deci mals, which never terminate.

EXAMPLES.

-

Thus 2/24X √2==√4X2=√8
And 23/4-3/8×3/4=3/8×4=3/32
Also 3a9Xa=√9Xa=9a
And /4a= X 3/4a=V/X4a=/c

1. Let 5/6 be reduced to a simple radical form.
2. Let 5a be reduced to a simple radical form.

2a

9

3. Let 2/ be reduced to a simple radical form.

3

4a2

'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 23 to quantities that shall have the

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2. Reduce 5 and 6 to quantities that shall have the

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common index

6

3. Reduce 2 and 4a

common index

1

8

to quantities that shall have the

4. Reduce a2 and a to quantities that shall have the

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

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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.

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

4. Reduce a common index.

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to quantities that shall have a

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. (r)

EXAMPLES.

1. Let 48 be reduced to its most simple form. Here 48/16X3=4/3 Ans.

2. Let /108 be reduced to its most simple form. Here 3/1083/27X4=33/4 Ans.

Note 1. When any number, or quantity, is prefixed to the surd, that quantity must be multiplied by the root of the factor above mentioned, and the product be then joined to the other part, as before.

EXAMPLES.

1. Let 2/32 be reduced to its most simple form. Here 2/32=2/16X2=8/2 Ans

2. Let 5 3/24 be reduced to its most simple form. Here 53/24-53/8X3=103/3 Ans.

Note 2. A fractional surd may also be reduced to a more convenient form, by multiplying both the numerator and denominator by such a number, or quantity, as will make the denominator a complete power of the kind

(r) When the given surd contains no factor that is an exact power of the kind required, it is already in its most simple form. Thus, 15 cannot be reduced lower, because neither of its factors, 5, nor 3, is a square.

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