082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

1.2 Laws of indices 9
Similarly
(2 1/3 ) 3 = 2 1 = 2
so that2 1/3 isacube rootof 2:
√
2 1/3 = 3 2=1.2599...
Ingeneral 2 1/n isannthrootof 2.The general lawstates
x 1/n isannthrootofx
Example1.10 Write the following usingasingle positive index:
(a) (3 −2 ) 1/4 (b) x 2/3 x 5/3 (c) yy −2/5 (d) √ k 3
Solution (a) (3 −2 ) 1/4 = 3 −2×1 4 = 3 −1/2 = 1
3 1/2
(b) x 2/3 x 5/3 =x 2/3+5/3 =x 7/3
(c) yy −2/5 =y 1 y −2/5 =y 1−2/5 =y 3/5
(d) √ k 3 = (k 3 ) 1/2 =k 3×1 2 =k
3/2
Example1.11 Evaluate
(a) 8 1/3 (b) 8 2/3 (c) 8 −1/3 (d) 8 −2/3 (e) 8 4/3
Solution We notethat 8 maybewritten as2 3 .
(a) 8 1/3 = (2 3 ) 1/3 = 2 1 = 2
(b) 8 2/3 = (8 1/3 ) 2 = 2 2 = 4
(c) 8 −1/3 = 1
8 1/3 = 1 2
(d) 8 −2/3 = 1
8 2/3 = 1 4
(e) 8 4/3 = (8 1/3 ) 4 = 2 4 = 16
Engineeringapplication1.4
Skindepthinaradialconductor
When an alternating current signal travels along a conductor, such as a copper wire,
most of the current is found near the surface of the conductor. Nearer to the centre
of the conductor, the current diminishes. The depth of penetration of the signal,
termed the skin depth, into the conductor depends on the frequency of the signal.
Skin depth, illustrated in Figure 1.1, is defined as the depth at which the current
➔