Answer: (a) [tex]e^3[/tex]
(b) [tex]e^{10}[/tex]
(c) [tex]\dfrac{1}{e^4}[/tex]
(d) [tex]e^{4}[/tex]
Step-by-step explanation:
Properties of exponents :
1) [tex]a^m\times a^n=a^{m+n}[/tex]
2) [tex]\dfrac{a^m}{a^n}=a^{m-n}[/tex]
3) [tex](a^m)^n=a^{mn}[/tex]
4) [tex]a^{-n}=\dfrac{1}{a^n}[/tex]
Now , we simplify the given expression using the above properties.
a) [tex](e^6)(e^{-3})[/tex]
[tex]=e^{6+(-3)}[/tex] [By property (1)]
[tex]=e^{6-3}=e^3[/tex] [∵ (+)(-)=(-)]
b) [tex](e^{-2})^{-5}[/tex]
[tex]=e^{-2\times-5}[/tex] [By property (3)]
[tex]=e^{10}[/tex] [∵ (-)(-)=(+)]
c) [tex](\dfrac{e^6}{e^2})^{-1}[/tex]
[tex]=(e^{6-2})^{-1}[/tex] [By property (2)]
[tex]=(e^{4})^{-1}[/tex]
[tex]=\dfrac{1}{e^4}[/tex] [By property (4)]
d) [tex](e^3)^{\frac{4}{3}}[/tex]
[tex]=e^{3\times\frac{4}{3}}[/tex] [By property (3)]
[tex]=e^{4}[/tex]
