2

Therefore (a) and (b3)3 or √/ao and √✅\/b3 are the quantities.

3. Reduce 43 and 5 to the common index 4, and 3-5 and 5-4 to the common index 2, and then to the common index -2.

Ans. (256) and 25*; (325) and (5-2)2;

4. Reduce a3 and to the common index . 5. Reduce a2 and a3 to the same radical sign. 6. Reduce (a + x)3 and (a

x) to a common index.

7. Reduce (a + b) 5 and (a — b)-25 to a common index.

I

3

Ans. (a) and (x2).

Ans.

a and √ x6.

8. Transform a―b-6c3d3 25 to another quantity whose index is

3.25.

PROBLEM III.

To reduce surds to simpler forms.

DIVIDE the surd, if possible, into two factors, one of which is a power of the kind that accords with the root sought; as a complete square, if it be a square root; a complete cube, if it be a cube root; and so on. Set the root of this complete power before the surd expression which indicates the root of the other factor; and the quantity is reduced as required.

If the surd be a fraction, the reduction is effected by multiplying both its numerator and denominator by some number that will transform the denominator into a complete square, cube, or other requisite power: its root will be the denominator to a fraction that will stand before the remaining part, or surd. See Ex. 3, below.

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.

3/3203 (64 × 5) =3/64 × 3/5 = 4 × √5 = 43/5. 3. Reduce to simpler terms.

√(15.5

44

4. Reduce √75 to its simplest terms.

5. Reduce 3/-189 to its simplest terms.

6. Reduce to its simplest terms.
7. Reduce ✓75a2b to its simplest terms.

Ans. +3310.

Ans. 5a/3b-1.

8. Express the square root of a2b3c1 in the simplest form.

Note I. There are other cases of reducing algebraic surds to simpler forms, that are practised on several occasions; one of which, on account of its simplicity and usefulness, may be here noticed, viz. in fractional forms, having compound surds in the denominator, multiply both numerator and denominator

by the same terms of the denominator, but having one sign changed, from + to or from to +, which will reduce the fraction to a rational denominator.

And the same method may easily be applied to examples with three or more surds.

Ex. 2. Reduce the fractions

√3
√3 + √5

X

√2 √18 √16

and

√2

4

to others having rational denominators: and then to such as have rational

numerators.

Note II. In the same manner may any binomial surd be rendered rational in the denominator, whatever the degree of the radicals may be. If, for instance, then the multiplier would be a2 + 3√ ab +3√ b2,

C

the surd had been a±3⁄4√b

and the surd itself become

And generally since

a+b

a ± b

"~/a ± √b="~/a" - 1 F "√a^-2 b + √a^-3 b2 + ........... by actual division, the rule may be extended as we have stated above.

1. BRING all fractions to a common denominator, and reduce the quantities to their simplest terms, as in the last problem. 2. Reduce also such quantities as have unlike indices to other equivalent ones, having a common index. 3. Then if the surd part be the same in them all, annex it to the sum of the rational parts, with the sign of multiplication, and it will give the total sum required.

But if the surd part be not the same in all the quantities, the addition can only be indicated by the signs + and

EXAMPLES.

1. Required to add √18 and √32 together.

First, 18=
= √9 × 2 = 3√2; and √32 = √16 × 2 = 4 √2.
Then, 3√2 + 4 √2 = (3 + 4) √/2 = 7 √2 = sum required.

2. It is required to add 3/375 and 3/192 together.

First, 3/375 =3/125 × 3 = 53⁄4/3; and 3/192 = 3/64 × 3 = 43√√/3.
Then, 5/3 + 43/3 = (5

and

√15

√8-√2 √5
√8+ √2 √5+ √15

Ans. - 3/7.
Ans. 3/2.

Ans. (3a+20a2) √b.

Ans. 5a or -11a.

:: also of a/8, b√18,

a√27, b√45, b/125, and a/147; and again of 2/32, 9.2435, 5.68", 17 × 543, 33/432, 3/128,√/1452, 3/1458, 3635, and 11√/1331.

PREPARE the quantities the same way as in the last rule; then subtract the rational parts, and to the remainder annex the common surd, for the difference of the surds required.

But if the quantities have no common surd, the subtraction can only be indicated by means of the sign

EXAMPLES.

1. To find the difference between √320 and √✅80.

First, √320 √√/64 × 5 = 8√√√5; and √80 = √16 x 5 = 4√/5.
Then, 85-4√√√5 = 4√√5 the difference sought.

2. To find the difference between 3/128 and 3/54.

First, 3/1283/64 x 2 = 43/2; and 3/54 = 3/27 x 2 = 33/2.
Then, 43/2 — 33/2 = 3/2, the difference required.

',

3. Required the difference of √75 and √ 48.

4. Required the difference of 3/256 and 3/32.

5. Required the difference of and √.

6. Find the difference of 3 and √.

7. Required the difference of / and /.

8. Find the difference of √24a+b2 and √54b1.

1

Ans. √3.

Ans. 23/4.

Ans. 3.

Ans. 16.

Ans. 3/75.

Ans. (3b2-2ab) √6.

9. Subtract 95 from 64-3 and add the sum to the difference of and

REDUCE the surds to the same index, if necessary; next multiply the rational quantities together, and the surds together; then annex the one product to the

other for the whole product required; which may be reduced to more simple terms if necessary.

EXAMPLES.

1. Required to find the product of 4/12 and 3/2.

Here 4 × 3 × √12 × √2 = = 12/12 x 2 = 12√24 = 12/4 × 6 = 12 x 2 × √6: =246, the product required.

2. Required to multiply / by 11.

8. Required the product of (a + b)' and (a + b)a.
9. Required the product of 2x + b and 2xb.

10. Required the product of — (a + 2 √/b), and (a — 2 √/b)3.

1

1

+

1 2√3

by

3√2-√3

•5

and/3.(√√5—1) by √/27 (√5+1);

15. Reduce (a)·2 × (±a)−5 × (±a)7 × (±a) to a single term.

PROBLEM VII.

To divide one surd quantity by another.

REDUCE the surds to the same index, if necessary; then take the quotient of the rational quantities, and annex it to the quotient of the surds, and it will give the whole quotient required; when the result can be reduced to more simple terms, it should then be done.

EXAMPLES.

1. Required to divide 6/96 by 38.

Here 63. √ (96 ÷ 8) = 2 √12 = 2 √ (4 × 3) = 2 x 2√3 = 4√ 3, the quotient required.

2. Required to divide 12 3/280 by 3 3/5.

Here 12 ÷ 3 = 4, and 3/56 = 3/8 × 3/7 = 2 3/7 ;

therefore 4 x 2 x 3/7 = = 8 3/7, is the quotient required.

3. Let 4/50 be divided by 2√5.

4. Let 63/100 be divided by 3 3/-5.

Ans. 2/10.

Ans. -23/20.

RAISE both the rational part and the surd part. Or multiply the index of the quantity by the index of the power to which it is to be raised, and to the result annex the power of the rational parts, which will give the power required.

EXAMPLES.

1. Required to find the square of a.

First, (3)2 = 4 × 3 = fs, and (a3)2 = a* ×2 = aa = a;

therefore (a3)2 = {a, is the square required.

2. Required to find the square of a3.

First, × =, and (a3)2

therefore (a3 =

= a3 = a3√a;

a3/a, is the square required.

3. Required to find the cube of √ 6 or 3 × 6.

First, (3)3 = × × }=, and (6)
3 3 1⁄2, and (64)3 = 6a — 6 √/ 6 ;

=

therefore (6)3 = 2× 6 √6 = 166, the cube required.

4. Required to find the square of 2 3/2.

5. Required the cube of 34,

or √3.

Ans. 4 3/4.

Ans. 3/3.

I. WHEN the given expression contains but one term, extract both the rational part and the surd part. Or divide the index of the given quantity by the index of the root to be extracted; then to the result annex the root of the rational part, which will give the root required.

therefore (16√/6) = 4.6 ± 44/6, is the square root required.

2. Required to find the cube root of

√3.

therefore (√/3) = 1.3/3, is the cube root required.