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If the surds are not in their simplest form, it will save labor to reduce them to this form before multiplication.

Example. The product of 5√32, √48, 2√54

=5.4√2×4√3×2.3√6=480.√2. √3. √6=480×6=2880. 264. To multiply surds of different orders: reduce them to equivalent surds of the same order, and proceed as before.

Example. Multiply 53/2 by 2√/5.

The product = 5 √/22 × 2 √/53 = 10/22 × 53=10/500.

DIVISION OF Surds.

265. Suppose it is required to find the numerical value of the quotient when 5 is divided by √7.

At first sight it would seem that we must find the square root of 5, which is 2.236..., and then the square root of 7, which is 2·645..., and finally divide 2.236... by 2·645...; three troublesome operations.

But we may avoid much of this labour by multiplying both numerator and denominator by 7, so as to make the denominator a rational quantity. Thus

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No5
√7

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266. The great utility of this artifice in calculating the numerical value of surd fractions suggests its convenience in the case of all surd fractions, even where numerical values are not aNo√√b Thus it is usual to simplify √c

required.

as follows:

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The process by which surds are removed from the denominator of any fraction is known as rationalizing the denominator. It is effected by multiplying both numerator and denominator by any factor which renders the denominator rational. We shall return to this point in Art. 270.

267. The division of surds then may be effected by expressing the result as a fraction, and rationalizing the denomi

nator.

16.

of

18.

Example 2.

22.

Example 1. Divide 4/75 by 25/56.

The quotient =

26.

Find the value of

1. 2/14x21. 2. 3/8 × √6.

4. 215 x 35. 5. 8/12x3/24.

7. 21/384÷8/98. 8. 5/27÷3√24.

10.

13. a/b3x b2 Ja.

4

14

√2

=

4/75 4 x 5/3 2/3
25/56 25 x 2/145/14
2/3x/14 2/42 √42
5/14x145x14 35

60

√5

1

√500

3/b/b x Je be be
$/c2
3/c3×3/c/c

с

EXAMPLES XXXII. c.

9.

168x/147. 11. 5128 × 2/432. 12. 6/14÷2/21.

14.

23

19.

27.

25

√5

23. 144÷÷/6.

4 √243

=

5
3/11
2/98 7/22

17.

x x

Given /2=1.41421, √3=1·73205, √5=2.23607, √6=2·44949, √7=264575: find to four places of decimals the numerical value

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3
a-b N

20.

28.

3.

6.

15.

5√ax 2√3.

x+2xx-2. 13/125÷5/65.

25

√252

10

√7°

24. √2÷√3. 25.

3/48 6/84 5/112/392

=

2.x

18x3

1-6 (a-b)5

21.

29.

48

√6

1

2/3

256

1575

268. Hitherto we have confined our attention to simple surds, such as 5, Ja, √x+y. An expression involving two or more simple surds is called a compound surd; thus 2√α-3√b; a+b are compound surds.

269. The multiplication of compound surds is performed like the multiplication of compound algebraical expressions.

Example 1. Multiply 2x-5 by 3x.

The product = 3√x (2√x − 5)

=6x-15/x.

Example 2. Multiply 2/5+3/x by √5-√x.
The product (2/5+3x) (√5-√x)

=2/5.√5+35.√x-2√5.√x-3√x.√x

=10-3x+√5x.

Example 3. Find the square of 2/x+ √7−4x.
(2√x+ √7 −4x)2= (2.√/x)2 + (√7 −4x)2+4√x.No√7-4x
=4x+7-4x+4/7x-4x2
=7+4/7x-4x2.

EXAMPLES XXXII. d.

Find the value of

1. (3x-5)×2√x.

3. (√a+√b) x√ab.

5. (2√3+3√2)2.

7. (3/5-4/2) (2.√5+3,√2). 9. (√x+√x−1) × √x − 1 . 11. (√a+x-2 Ja)2.

13. (a+x-Na-x)2.

15. (√2+√3-√5)(√2+√3+√5).
16. (√5+3/2+√7)(√5+3√√2-√7).

N

Write down the square of

17. √2x+a-√2x-a.

19. √m+n+√m-n.

21. 3x√2-3/7-2x2.

2. (√x-a)x2x. 4. (√x+y-1)x√x+y. 6. (√7+5√3) (2√7 −4,√3). 8. (3a-2x) (2√a+3√↓x). 10. (√x+a−√x− a)×√x+a. 12. (2/a-√1+4a). 14. (√a+x-2)(√a+x−1).

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270. One case of the multiplication of compound surds deserves careful attention. For if we multiply together the sum and the difference of any two quadratic surds we obtain a rational product.

Examples. (1) (√a+√b) (Ja−√b) = (√a)2 - (√√b)2=a-b.

(2) (3√5+4√3) (3.√5 – 4,√3) = (3/5)2 — (4, √/3)2=45-48=-3. Similarly, (4 – √√a+b) (4 + √a+b)=(4)2 − ( √a + b)2 = 16 ~ a − b.

271. DEFINITION. When two binomial quadratic surds differ only in the sign which connects their terms they are said to be conjugate.

Thus 3/7+5/11 is conjugate to 3/7-5/11.
Similarly, a-a-2 is conjugate to a+√a-x2.

The product of two conjugate surds is rational. [Art. 270.]
Example. (3/a+ √x-9a) (3√a - √x-9a)

=(3√a)2-(√x-9a)2-9a-(x-9)=18α-x.

DIVISION OF COMPOUND SURDS.

272. If the divisor is a binomial quadratic surd, express the division by means of a fraction, and rationalize the denominator by multiplying numerator and denominator by the surd which is conjugate to the divisor.

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The expression

4+3/24+3/25+3/2
5-3/2 5-3/25+3/2

X

20+18+12/2+15√2 _ 38+27√2

=

25-18

7

Example 2. Rationalize the denominator of

b2

α

√a2+b2 - a
√a2+b2+a^ √a2+b2-a
b2a2+b2-a}
(a2+b2) - a2

= √a2 + b2 − a.

==

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b2

√a2+b2+a

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1

2+/3

=2/3, on rationalizing.

by

√3+√2 7+4√3
2-3 √3-√2
√3+√2√3-√2
2-3 7+4/3
(√3)2- (√2)
14-12+8/3-7/3

X

Example 4. Given √5=2.236068, find the value of

The expression:

Rationalizing the denominator,

=

It will be seen that by rationalizing the denominator we have avoided the use of a divisor consisting of 7 figures.

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87 87 (7+2/5)
7--2/5 49-20
=3(7+2/5)

=34.416408.

273. In a similar manner, where the denominator involves three quadratic surds, we may by two operations render that denominator rational.

Example.

=

=

=

√2(√2+√3+ √5)

(√2+ √3−√5)(√2+√3+√5)

2+√6+ √10

2√6

87

7- 2No√√5°

(2+ √6+√10) √6
(2√6) √6

3+√6+ √15

6

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