Understanding Engineering Mathematics

4.2.4 Manipulation of the exponential function

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119 136➤

As noted earlier, any exponential function, includingex, satisfies the usual rules of
indices (18
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):

exey=ex+y

ex

ey

=ex−y

(ex)y=exy

e−x=

1

ex

e^0 = 1

You need to be very proficient at using these rules. You will frequently need to manipulate
functions of exponential functions, whatever area of engineering you enter.
Perhaps the most important thing to remember about the exponential function is that
eA+Bisnotequal toeA+eB. This is an error made by many beginners. The correct result
is of course

eA+B=eAeB

Solution to review question 4.1.4

A. (i)eAeB=eA+B (ii)eA/e−B=eAeB=eA+B

(iii) e^2 B(e^3 B)^2 =e^2 Be^6 B=e^8 B

(iv)

eAe^2 Be−C

e^2 AeBeC

=e−AeBe−^2 C=e−A+B−^2 C

(v)e−Be−CeB=e−Cusinge−BeB= 1

(vi)(eA)^3 e−^2 A=e^3 Ae−^2 A=eA

B. (i)

e^2 A−e^2 B

eA+eB

=

(eA)^2 −(eB)^2

eA+eB

=

(eA−eB)(eA+eB)

eA+eB

=eA−eB

(ii) (eA−e−A)(eA+e−A)=(eA)^2 −(e−A)^2

=e^2 A−e−^2 A

(iii)

e^2 A+ 1

e^2 A+e−^2 A+ 2

=

e^2 A+ 1

(eA)^2 + 2 +(e−A)^2

=

e^2 A+ 1

(eA+e−A)^2