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

4 Chapter 1 Review of algebraic techniques
Note that in this formula forP, the voltage is raised to the power 2. Note an important
consequence of this formula is that doubling the voltage, while keeping the
resistance fixed, results in the power dissipation increasing by a factor of 4, that is
2 2 .Alsotreblingthevoltage,forafixedvalueofresistance,resultsinthepowerdissipation
increasing by a factor of 9,thatis3 2 .
Similar considerations can be applied to Equation 1.1. For a fixed value of resistance,
doubling the current results in the power dissipation increasing by a factor of
4,andtreblingthecurrentresultsinthepowerdissipationincreasingbyafactorof9.
Consider the product 3(3 3 ). Now
3(3 3 ) = 3(3.3.3) = 3 4
Also, using the first law of indices we see that 3 1 3 3 = 3 4 . This suggests that 3 is the
same as 3 1 . This illustratesthe general rule:
a=a 1
Raising a number tothe power 1 leaves the number unchanged.
Example1.4 Simplify (a) 5 6 5 (b) x 3 xx 2
Solution (a) 5 6 5 = 5 6+1 = 5 7 (b) x 3 xx 2 =x 3+1+2 =x 6
1.2.2 Dividingexpressionsinvolvingindices
Consider the expression 45
4 3:
4 5
4 = 4.4.4.4.4
3 4.4.4
= 4.4 by cancelling 4s
= 4 2
This serves toillustratethesecond lawof indiceswhich is
a m
a n =am−n
When expressions with the same baseare divided, the indices aresubtracted.
Example1.5 Simplify
(a) 59
5 7 (b) (−2)16
(−2) 13
(c) x9
x 5
(d) y6
y
Solution (a)
(b)
5 9
5 = 7 59−7 = 5 2
(−2) 16
(−2) = 13 (−2)16−13 = (−2) 3