Final answer:
To solve the equation 3^n = 3^x / 9^y for n, rewrite 9^y as (3^2)^y and use the properties of exponents to simplify. The bases are the same, so set the exponents equal to each other to find that n = x - 2y.
Step-by-step explanation:
To find an expression for n in terms of x and y in the equation 3n = 3x / 9y, we need to use the properties of exponents. Remember that 9 is 3 squared, so we can rewrite the denominator as (32)y. This gives us:
3n = 3x / (32)y
Using the power of a power rule for exponents, which states that (ab)c = abc, we simplify the denominator further:
3n = 3x / 32y
Now, using the quotient rule for exponents, which states that am / an = am-n, we get:
3n = 3x - 2y
Since the bases are the same and the equation is only true if the exponents are equal, we can deduce that:
n = x - 2y