Answer:
[tex]n=30[/tex]
Step-by-step explanation:
Given expression:
[tex]3^1\times 9^2\times 27^3\times 81^4[/tex]
The given expression needs to be simplified to the form [tex]3^n[/tex]
Applying the exponents rule to simplify.
[tex]3^1\times 9^2\times 27^3\times 81^4[/tex]
Writing all numbers as product of 3.
⇒ [tex]3^1\times (3\times3)^2\times (3\times3\times3)^3\times (3\times3\times3\times3)^4[/tex]
Writing each in exponents form.
⇒ [tex]3^1\times (3^2)^2\times (3^3)^3\times (3^4)^4[/tex]
Using power of a power rule [tex][\ (a^x)^y=a^{xy}\ ][/tex]
⇒ [tex]3^1\times 3^4\times 3^9\times 3^16[/tex]
Using multiplication rule of exponents [tex][\ a^x\times a^y=a^{x+y}\ ][/tex]
⇒ [tex]3^{1+4+9+16}[/tex]
⇒ [tex]3^{30}[/tex]
So we have [tex]3^n=3^{30}[/tex]
∴ [tex]n=30[/tex]
