TRACT 10.

FOR EXTRACTING ROOTS.

211

then as 748-2

748-3

: 1 :

7483 _ , t7482 “ 1tttt

= 1-00013366,

the root sought, very exact at one operation.

2 a

15. Required to find the value of the quantity (5 a) t or (y ) T .Now this may be done two ways; either by finding thepower or | root of y at once ; or else by finding the 3d orcubic root of A?, and then squaring the result.

By the first way:—Here it is easy to see that a is nearly

3

= 3, because 3 T =v/27 = 5 + some small fraction. Hence,to find nearly the square root of 27, or V 27, the nearestpower to which is 25 = a z in this case :

Hence 3 n + a* = 3 x 27 + 25 = 106,and n + 3a ! = 27 + 3 X 25 = 102 ;

5 x 53 265

then 102 : 106, or 51 : 53 : : 5 : -— —• — y/27 nearly,

51 51

Then having n =

21 3 A 265

«-= -j, a-3, anda l =— nearly;

it will be |n +

and an + |

5 21 1

i^T+T*

s

a* =

21

X — + ■

265 _ 6415IT - 408 ’265 _ 6371TT~ 408 ’

hence 6371 : 6415 :: 3: = 3y T yV = 3-020719, for the

value of the quantity sought nearly, by this way.

Again, by the other method, in finding first the value of

I

(y) T , or the cube root of y. It is evident that 2 is the

nearest integer root, being the cube root of 8 = a 3 .

Hence 2 n + a 1 = y + S = y,

and N + 2« 3 = y 16 = y ; '

148 7 7

then 85 : 74 : : 2:or =—nearly. Then taking — fora,

, ,21, 343

we have 2 n + a 3 = —-4—— =2 64

1015

64~’

and n + 2a J =

2.343

102264 ’

64