Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

.

84

Given 20-32=5; and Given 28 + 9x=30, to find 2.
0
-3

9

0 ....0 5(3.4259887573622 .... 4 30 (2.1808498 0 0

Div. 13

26
9
5

4 4 6.4..2.56 4.624

6.1.... 61 2.161 Div. 11.56 376

Div. 21.61 1.839 16 288488

1 1.819232 7.22...1444 87512 6.38....6104

19768 Div. 14.4244 73027625 Div. 22.7404 18610 4 14484375

64

1158
7.265.. 3633
13183581
6.54 .... 52

931
14.60553
1300794

23.2624

227 7/21759...655 1172444

2|3.217

209 14.6484|2 841 128350

18 57 733 117250

Root,

18
14.655|5 108 11100

x=2.1808498
14.6|5|6|1 103 10259
.

Givenx+2+=100 find %. Given 2 12x = 12, to find x.

1 -12 -12(1.11574951 5.... 20

100 (4.2644 1.... 1 --11

676 Div. 21 - -1

16 16 869

1.32 .. 2.64 11.9 28 3.1.... .31 - -131

Div. 59.64 4.072 Div. 8.69 83369

04 3.788376 1 --47631 16.66...8196

283624 3.31

331
-41435125

Div. 63.1396 256072 Div. 8.3369 -6195875

36

27552 1

55136 -5787588

25631 3.335.... 16675

408287
Div. 64.017936

1921 Div. -8.287025 -330625

1282 25

13.792.. 5517 -78662

639 2342 --74390 64.078605

577 8.267983

276 56-4272

62 13

64.078881
50-4133
8.26563

1
6 -139
-8.261516

83
6/4.0/719

4 31. Two travellers, A and B, set out together from the same place ; A travels 8 miles the first day, 12 the second, 16 the third, &c.; B

goes 1 mile the first day, 4 the second, 9 the third, &c. ; how many days must they travel before B overtakes A.

7 days.

-11

[ocr errors]

13.784...

16

3.3457....

[ocr errors]

of a.

[ocr errors]

Of Irrational Quantities, of Surds. Irrational Quantities, or Surds, are those of which the values cannot be accurately expressed in numbers, or are such as have no exact root, and are usually expressed by means of the radical sign Ķ, 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, 1/2, or 27, denotes the square root of 2; 2Na, or at the cube of the square of a, and am, is the mtb Va,

a, root of the nth

power 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 to which it is to be reduced ; and over this new quantity place the radical sign, or proper index, and it will be of the form required. 1. Let 3 and 5 be reduced to the form of the square root.

Here 3X3=3=9; whence 79. Ans,

Here (5)=25; therefore 725, Ans. 2. Reduce 2.2 and 3x to the form of the cube root.

Here (2x):8–876; whence w/8x®, or (82935

Here (32)_2728; therefore 9-27x, Ans. 5. Let —2a be reduced to the form of the fourth root.

Here (2a)=16a"; therefore W16a", Ans. 6. Let a be reduced to the form of the fifth root, and wat

wa NE,

and to the form of the square root. 2a

bna Here (a) =a"; therefore a", Ans. And (watozatb+2wab; :: W(a+b+2Nab). Again,

1 = 2a

; . Nda) 42:(ła). Also, wala pa i no

77 ° * A quantity of the kind here mentioned, as for instance 2, is called an irrational number, or a surd, because no number, either whole or fractional, can be found which, when multiplied by itself, will produce 2. But its approximate value may be determined to any degree of exactness by the common rule for exfracting the square root, being 1 and certain non-periodic decimals, which never terminate.

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.

Thus 2x2=N4XNN (4x2)=18,
And 2948X45 (8X4)=W32.
Also 319XNN 19Xa)=9a,
And 4a=»IX 34a=N(3X4a)=wS

[ocr errors]

a

a

4a

as

2a

=w10=3

4a?

.

[ocr errors]

9 8q 9

72a 72a 2a Here

ラージラ×
3
27

108a
1. Reduce 576 and 5a each to a simple radical form.

Here 5/6=W25XW6= 150, Ans.

Here $5a=N 25X50=N(), Ans. 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 she 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. 1. Reduce

31 ,
and 23, or

1
61
or 52 and 6

or aand 6

si

each to quantities that shall have the index .

Here 1:5X=33, the first index ;

And +=X=1=2, the 22 index. Whence (89) * and (2) ¢, or 276, and 46, are the quantities required.

. Here 1 + $=x=3, 1st index, $=1= x=2, 2d index ; herce (59and (67!, or 125% and 366, are the answers sought: )

': Here ixt=3, 1st index; =fx=4, 2d index. Therefore (a) and ()# are the answers.

3. Reduce 21 and 41, to quantities that shall have the common index $

Here + sixt=4, 1st index, -=X=2, 2d index; therefore (24# and (43_165 and 16% $

Ans. 166 and 165 4. Reduce c2 and a to quantities that shall have the common index 1. Here 4+1=x128, 1st index ; 1=1=X12, 2d.

子 Therefore (29) and (a?) are the quantities sought. 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 e, power denoted by its nume1. Reduce 34 and 4 or 4 and 54, or it and at, or at and !

12 each to quantities having a common index. Here 31–38=(39)# =(39)&=(27), and 4

43=4&=(49)+=(1

42) Whenice (27;&and (16) +, Ans. Here 1 and 1 = t; and y's; hence (46) and (5)*, Ans. $

) 59

[ocr errors]
[ocr errors]
[ocr errors]

=4 =

[ocr errors]
[ocr errors]

and (6975,

[ocr errors]

m

n

n m

mn

m n

I

[ocr errors]
[ocr errors]

Here f=fand f=8; therefore ()6 and (, Ans.

I'm Here again j=ih and = id; therefore (a*)

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

1 1 Here reduces to the common denominators and

; therefore (am)

and (6")"
(60

Ans.
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.*

1. Let 48 and „108 each be reduced to its most simple form. Here 48=(16X3)=473, and 108=(27X4)=374.

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 then joined to the other part, as before. 1. Let 2X32 and 53/24 each be reduced to its most simple form. Here 2N32=2N (16X2)=812, & 5724—5 (8X3)=10%/3.

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 required ; and then joining its root, with 1 put over it, as a numerator to the other part of the surd.

1 1. Let Vy, 125, 294, 56, V192, 7/80, 981, and TÄIN, each be reduced to its most simple form.

Here „Ž=N=N(X14)=1N14, Ans. 3733 =//GX50)=50, Ans. „125 (25x5)=575, Ans.

294=N(49 X6)=75, Ans. 356=(8X7)=277, Ans. 192=64X3)=4/3, A. 7/80=7V (16x5)=2875, A. 9781= (27X3)=27/3, Ans.

Here, reducing the radical, we have v = = $30; therefore TINTIX/30=zx/30, Ans.

10. Reduce 1698aRx, and (2_a*x*), each to the most simple form. Here Ni=No=112 ;; hence wi=112, Ans.

Ņ98a+z=(49a2x2x)=7ax/2x, Ans.

(2-a)={x*(x--a?)}=XN (a), Ans. *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, N15 cannot be reduced lower, because neither of its factors, 5 nor 3, is a square.

12

64

CASE IV. To add Surd Quantities together.

RULE. When the surds are of the same kind, reduce them to their simplest forms, as in the last case; then, if the surd part be the same in each of them, annex it to the sum of the rational parts, and it will give the whole sum required.

But if the quantities have different indices, or the surd part in each be not the same, they can only be added together by the signs

+ and

3N

1. It is required to find the sum of „27 and „48.

27=)=33 And 74&V (163= }=713 the sum.

2. It is required to find the sum of 500 and 108.
Here 500=(125X4)=594)
And 108W 27X4)=34)

A} .894 the sum.
3. Required to find the sum of 4N 147 and 3x75.
Here 4w 147=41 (49X3=2873
And 375 33V 25X3=151}}=439/3 the sum.

()3
4. Required to find the sum of 31 and 2N to.
Here 34=3N4=10

- İN
And 2 =28= 10 } =$v10 the sum.
2Ni2Nikon

Vio
5. Required to find the sum of 772 and 7128.
First Ni2=^ (36X2)=672
And V128=> 64x2=872}=147/2, Ans

.
()
6. Required to find the sum of „180 and „405.

=) Also V405=V81X5=975}=1575, Ans.

=) 7. Required to find the sum of 3%40 and 135. First 30=3(8x5)=635 And 135=(27x5)=335)

5}995

=9/5, Ans. 8. Find the sum of 454 and 5128. Here 4/54347 (27X2)=122 And 5128=5764X2)=2092)

... 3272, Ans. 9. Find the sum of 9/243 and 10/363.

9 = 97) = And 10.7363–10/(121*3=11073}=191V/3, Ans.

) 10. Find the sum of 37 and 71. By first reducing the fractional surds, we have

N}=N=16, and N=N=ion (9X6)=*N6; Hence 33 X 116=16

= } And I=7X16=476)

VE}=w6, Ans.

iI
11. Find the sum of 12 and 3 y.
Here N=N=}/2, and 2;
Hence 1271+3=67/2+122=67/2, Ans.
12. Find the sum of twa?b and 462*.

1

« ΠροηγούμενηΣυνέχεια »