To find the Square root of a Compound Quantity.

THIS is performed like as in numbers, thus:

1. Range the quantities according to the dimensions of one of the letters, and set the root of the first term in the quotient.

2. Subtract the square of the root, thus found, from the first term, and bring down the next two terms to the remainder for a dividend; and take double the root for a divisor.

3. Divide the dividend by the divisor, and annex the result both to the quotient and to the divisor.

4. Multiply the divisor thus increased, by the term last set in the quotient, and subtract the product from the dividend.

And so on, always the same, as in common arithmetic.

EXAMPLES.

1. Extract the square root of a -4a3b+6a2b2 −4ab3+b*. a1-4a3b+a3b2 — 4ab3+b1 (a2 - 2ab+b2 the root.

2a* - 2ab) - 4a3b+6a2b2

-4a3b+4a3b2

2a2-4ab+b2) 2α2 b2-4ab3+b1

2a3b2-4ab3+b$

2. Find

2. Find the root of a4+4a3b+10a2b2+12ab3+b1. a+4a3b+10a2b2+12ab3+9b* (a2+2ab+3b*.

as

2a3+2ab) 4a3b+10ab2

4a b+ 4a2b3

2a2+4ab+362) 6a2b2+12ab3+9bs
6a2b2+12ab3+9ba

3. To find the square root of a1+4a3+6a2+4a+1.

Ans. a2+2a+1.

4. Extract the square root of a1 ·2α3+2α3 a+1.

Ans +-+

5. It is required to find the square root of a2 -ab.

Ans. a--

b 63 b3
2 8a 16a3

&c.

CASE III.

To find the Roots of any Powers in General.

THIS is also done like the same roots in numbers, thus: Find the root of the first term, and set it in the quotient. Subtract its power from that term, and bring down the second term for a dividend.-Involve the root, last found, to the next lower power, and multiply it by the index of the given power, for a divisor.-Divide the dividend by the divisor, and set the quotient as the next term of the root.Involve now the whole root to the power to be extracted ; then subtract the power thus arising from the given power, and divide the first term of the remainder by the divisor first found; and so on till the whole is finished.*

EXAMPLES.

As this method, in high powers, may be thought too laborious, it will not be improper to observe, that the roots of compound quantities may sometimes be easily discovered, thus:

Extract the roots of some of the most simple terms, and connect them together by the signor, as may be judged most suitable for the purpose-Involve the compound root, thus found, to the proper power; then, if this be the same with the given quantity, it is the root required.-But if it be found to differ only in some of the signs, change them from+to, or from to, till its power agrees with the given one throughout.

Thus

EXAMPLES.

1. To find the square root of a1-2α3b+3a2b3-2ab3+bs. a*—2a3b+3a2ba — 2ab3+ba (a2 —ab+b2

■*—2a3b+3a2b3 — 2ab3+b1=(a3 —ab+b3)2.

2. Find the cube root of a6 - 6a5 +21α4 - 44a3 +63αa -54a+27.

a6a5+21a-44a3+63a2-54a+27(a2-2a+3.

a-6x5+21a4-44a3+63a2-54a+27=(a2-2-3)3,

Ans. a-b+x.

3. To find the square root of a2. 2ab+2ax+b2 — 2bx +x2. 4. Find the cube root of a63a59a4-13a3+18a212a+8. Ans. a2-a+2.

5. Find the 4th root of 81a4-216a3b+216a2 b2 - 96ab3 +166. Ans. 3a-2b. 6. Find the 5th root of a5 - 10a++ 40a3 — 80a2 + 80% -32. Ans. a-2.

7. Required the square root of 1-x2. 8. Required the cube root of 1-x3.

Thus, in the 5th example, the root 3a-26, is the difference of the roots of the first and last terms; and in the third example, the root a-bx, is the sum of the roots of the 1st, 4th, and 6th terms. The same may also be observed of the 6th example, where the root is found from the first and last tertns.

SURDS.

!

SURDS.

SURDS are such quantities as have not exact values in numbers; and are usually expressed by fractional indices, or by means of the radical sign✓. Thus, 3, or ✔✅ 3, denotes the square root of 3; and 23 or 3/22, or 3/4, the cube root of the square of 2; where the numerator shows the power to which the quantity is to be raised, and the denominator its root.

PROBLEM I.

To Reduce a Rational Quantity to the Form of a Surd.

RAISE the given quantity to the power denoted by the index of the surd then over or before this new quantity set the radical sign, and it will be the form required.

:

EXAMPLES.

1. To reduce 4 to the form of the square root.
First, 43=4X4-16; then 16 is the answer.
2. To reduce 3a2 to the form of the cube root.
First 3a2 X3a2 X3a2=(3a2)3=27a° ;

then 3/27a or (27a) is the answer.
3. Reduce 6 to the form of the cube root.

Ans. (216) or /216.

4. Reduce lab to the form of the square root.

Ans. ✔ab1.

5. Reduce 2 to the form of the 4th root.
6. Reduce a to the form of the 5th root.
7. Reduce a+x to the form of the square root.
8. Reduce a-x to the form of the cube root.

PROBLEM IL

Ans. (16)

To Reduce Quantities to a Common Index.

1. REDUCE the indices of the given quantities to a common denominator, and involve each of them to the power denoted by its numerator; then I set over the common denominator will form the common index. Or,

2. If the common index be given, divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities. Then over the said quan. tities, with their new indices, set the given index, and they will make the equivalent quantities sought.

EXAMPLES.

1. Reduce 3 and 53 to a common index. Here and

and 2.

Therefore 31% and 5% = (35)10 and (52)1'0 = 35 and

2. Reduce a and b to the same common index. Here.x the 1st index,

the 2d index.

10

Therefore (a) and (b*)3, or ✔ ao and √b3 are the quan

tities.

3. Reduce 43 and 5. to the common index 1.

Ans. 256) and 25.

4. Reduce a and

to the common index .

Ans. (a2) and (x2)*.

Ans. ✔a and √x".

5. Reduce a3 and x3 to the same radical sign.

6. Reduce (a+x) and (a-x) to a common index. 7. Reduce (a+b) and (a-b) to a common index.

PROBLEM III.

To Reduce Surds to more Simple Terms.

FIND out the greatest power contained in, or to divide the given surd; take its root, and set it before the quotient or the remaining quantities, with the proper radical sign between them.

EXAMPLES.

1. To reduce 32 to simpler terms.

Here/32/16×2=✓/16 × √2=4× √2=4√2. 2. To reduce 3/320 to simpler terms.

1/320=1/64×5=1/64 × V/5=4×3/5=43/5.

3. Reduce