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

2 Chapter 1 Review of algebraic techniques
Apart from being able to perform basic mathematical operations with vectors and
matrices, MATLAB ® has, in addition, a vast range of built-in computational functions
whicharestraightforwardtousebutneverthelessareverypowerful.Manyofthesehighlevelfunctionsareaccessiblebypassingdatatothemintheformofvectorsandmatrices.
A small number of these special functions are used and explained in this text. However,
to get the most out of a technical computing language it is necessary to develop
a good understanding of what the software can do and to make regular reference to the
manual.
1.2 LAWSOFINDICES
Considertheproduct6 ×6×6×6×6.Thismaybewrittenmorecompactlyas6 5 .We
call5theindexorpower. Thebaseis6.Similarly,y ×y×y×ymaybewrittenasy 4 .
Here the base isyand the index is4.
Example1.1 Write the following using index notation:
(a) (−2)(−2)(−2) (b) 4.4.4.5.5 (c)
Solution (a) (−2)(−2)(−2) may bewritten as (−2) 3 .
(b) 4.4.4.5.5 may be writtenas 4 3 5 2 .
yyy
xxxx
(d) aa(−a)(−a)
bb(−b)
(c)
yyy y3
may bewritten as
xxxx x 4.
(d) Note that (−a)(−a) = aa since the product of two negative quantities is positive.
Soaa(−a)(−a) =aaaa =a 4 . Alsobb(−b) = −bbb = −b 3 . Hence
aa(−a)(−a)
bb(−b)
= a4
−b = 3 −a4 b 3
Example1.2 Evaluate
(a) 7 3 (b) (−3) 3 (c) 2 3 (−3) 4
Solution (a) 7 3 = 7.7.7 = 343
(b) (−3) 3 = (−3)(−3)(−3) = −27
(c) 2 3 (−3) 4 = 8(81) = 648
Most scientific calculators have an x y button to enable easy calculation of expressions
ofasimilarform tothoseinExample 1.2.
1.2.1 Multiplyingexpressionsinvolvingindices
Consider the product (6 2 )(6 3 ). We maywrite thisas
(6 2 )(6 3 ) = (6.6)(6.6.6) = 6 5