For this case we have the following expression:
[tex]\frac {216 ^ {n-2}} {(\frac {1} {36}) ^ {3n}} = 216[/tex]
We multiply both sides by: [tex](\frac {1} {36}) ^ {3n}[/tex]
[tex]216 ^ {n-2} = 216 * (\frac {1} {36}) ^ {3n}[/tex]
We divide both sides by 216:
[tex]\frac {216 ^ {n-2}} {216} = (\frac {1} {36}) ^ {3n}[/tex]
To divide powers of the same base, we place the same base and subtract the exponents:
[tex]216 ^ {n-2-1} = (\frac {1} {36}) ^ {3n}\\216 ^ {n-3} = (\frac {1} {36}) ^ {3n}[/tex]
Rewriting:
[tex](6 ^ 3) ^ {n-3} = (\frac {1} {6 ^ 2}) ^ {3n}\\6 ^ {3n-9} = \frac {1} {6 ^ {6n}}\\6^{ 3n-9} * 6^{ 6n} = 1[/tex]
To multiply powers of the same base, we place the same base and add the exponents:
[tex]6^{ 3n-9 + 6n} = 1\\6^{ 9n-9} = 1[/tex]
We know that any number raised to zero is 1, [tex]a ^ 0 = 1.[/tex]
So, for equality to be true:
[tex]9n-9 = 0\\9n = 9\\n = \frac {9} {9}\\n = 1[/tex]
Answer:
[tex]n = 1[/tex]
