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2. Reduce 31 and 54 to surds of the same index.

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and /

į, reduced to a common denominator, arc

3

and . 6

Now 38=33=781 ; and 58=35*=125. Ex. 3. Reduce a at

Ans. Va and þes

to Surds Ex. 4. Reduce ct and di with the Ans. $cs and 4d.

do Ex. 5. Reduce 3 32&2 75

Ans. 3 4 & 2 7125.

index. Ex. 6. Reduce 41 and 15

Ans. 256 & 73375.

and 15 )

6

same

6

6

CASE III.

(131.) Surd quantities are reduced to their simplest form, by observing whether the quantity under the radical sign contains a power corresponding to the given surd root, and then extracting that root.

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Note.—The quantity without the radical sign is called the co-efficient of the surd; and it is evident, that this quantity may always be put under the radi. cal sigo, by rgieinw it on the power denoted by the index of the *****

Thus, 7a 72x=(by Case I.) v 7a x 7a x 2.r.

49ao xv2r=98aRx. Also, -x=wX* X v2a-.

=> x* X (2a- x)=v2ax?.

CASE IV. (132.) If the quantity under the radical sign be a fraction, it may be reduced to an integral form by the following

Rule.—Multiply the numerator and the denominator of the fraction by such a quantity as will make the denominator a complete power, corresponding to the root ; then extract the root of the fraction whose numerator and denominator are complete powers, and take it from under the radical sign.

Examples.

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The Fundamental Rules of Arithmetic applied to

Surd Quantities.

(193.) The Addition and Subtraction of Surd Quantities.

Rule.-Reduce the quantities to their simplest form; and if the surd part be the same in both, then their sum or difference will be found by taking the sum or difference of their co-efficients, and annexing the common surd to the result.

Examples. 1. Find the sum and difference of 16a®x and 74a®z. By Case III. 16aRx = 4a xx,

and ✓ 4aor = 2a vX; .. the sum =4a vr+2a 7x=(4a+2a) x vr=6a7r. the difference =40wx-2a7r=(42–2a) * vx=2a w*.

2. Find the sum and difference of 3/ 192 and 3/24.
By Case III. 192= 3/64 x 3 = 433,

and 3 245 38 X 3 = 233;
.. 3192+ 324=(4+2) X 33=63/3 or 233.

3. Find the sum and difference of Van

VV and 2, reduced to a common denominator,

8 The two fractions

27

48 27 are

and 162 102

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Note.-If the surd part is not the same in the qnantitics which are to be added or sabtracted from each other, it is evident thai the addition or sabtraction can only be performed by placing the sigus + or between them. 4. Add 27a*x and ✓3a*r together..

Ans. 4ao „3x. 5. ✓128 and 772

1472. 6. 3135 and 340

.535

7 7. Subtract 3

from 4

2, 8. 108 from 934....

...634. 9. Required the sum of 2 48 and 9 3 108.... Ans. 8/3+27 34. 10. Find the difference of į vi and in

Ans. 0. 11. Required the difference of 12x”ye and 27y*.

Ans. (2xy+3y)./3.

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V15.

(134.) The Multiplication and Division of Surd quantities.

Rule.- Reduce the quantities to equivalent ones with the same index, and then multiply or divide both the rational and the irrational parts by each other respectively.

Examples.

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1. Multiply va by 3/b, or aí by vs. The fractions and 3. reduced to common denominators, are dan

:: ai=aš=va'; and bú=b8=26. Hence vax Ml=Pax =?. 2. Multiply 5/5 by 3 8. 5/5x3/8=15/40=1524 X 10.

= 15 X 2 X v10=30/10. 3. Multiply 213 by 33 4. By reduction, 213=2x38=2x V3?=2227,

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and 334=3 X 43=3 y 34=3716 Hence 2 13 x 3 34=3 727X3716=6432 4. Divide 23 lc by 3 vac

27bc and 3 vac=3 x (ac)ă =3 Na'co;

23 bc_2 6 / 12c2 2 6 / 19

3 ac 3 a3c3=3 aic 5. Divide 103 108 by 5 3 4. Now 103 108=10* 27 *4=13X3 X 34=3034; 107 108 30 34

103108 = 6; or

2327=2x3=6 534 534

534 6. Multiply 315 by ✓10

Ans. 225000

Х

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(135.) On the Involution and Evolution of Surd quantities.

Rule.- Raise the rational part to the power or root required, and then multiply the fractional index of the surd part by the index of that

power or root.

Examples.

1. The square of Yaza;*2=0=80%. 2. The cube of 'b=b*3 = =6. 3. 4th power of 2.72=16x2; *4=16x2; =163 16=32 32. 4. Square root of a; b) =a} * 6*1 = 1

1

Х

5. Cube root of

6. Find the 4th power of į v2...... 7. Find the square of 72–75...... 8 Square 35..

.... Ans. Per ..Ans. 7-2_10.

Ans. 3/25.

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