Answer:
m = 9
Step-by-step explanation:
Using the rules of exponents
[tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]
[tex]a^{m}[/tex] × [tex]a^{n}[/tex] ⇔ [tex]a^{(m+n)}[/tex]
[tex]a^{-m}[/tex] = [tex]\frac{1}{a^{m} }[/tex]
Then
[tex](\frac{1}{27}) ^{m}[/tex] × [tex](81)^{-1}[/tex] = 243 ← express all values to base 3
[tex](\frac{1}{3^3}) ^{m}[/tex] × [tex](3 ^4)^{-1}[/tex] = [tex]3^{5}[/tex]
[tex](3^-3)^{m}[/tex] × [tex]3^{-4}[/tex] = [tex]3^{5}[/tex]
[tex]3^{-3m}[/tex] × [tex]3^{-4}[/tex] = [tex]3^{5}[/tex]
[tex]3^{(-3m-4)}[/tex] = [tex]3^{5}[/tex]
Since bases on both sides are equal, both 3 , then equate exponents
- 3m - 4 = 5 ( add 4 to both sides )
- 3m = 9 ( divide both sides by - 3 )
m = - 3
