Final answer:
The expression equivalent to (9^-3/3^-2 · 9^2)^3 is simplified by manipulating exponents, resulting in -(3^6/9^15) as the correct answer.
Step-by-step explanation:
To find the expression equivalent to the fraction (9^-3/3^-2 · 9^2)^3, we first simplify the expression inside the parentheses and then apply the exponent outside the parentheses.
Inside the parentheses:
9^-3 means 1/(9^3).
3^-2 means 1/(3^2).
9^2 is just 9 squared or 81.
Now we combine these expressions:
(1/(9^3)) ÷ (1/(3^2) · 81)
Simplifying further, we see that 9^3 is (3^2)^3 which is 3^6 and we can cancel out 81 with one of the 3^2 to get 3^4 in the denominator:
(1/3^6) ÷ (1/3^4)
We invert the second fraction and multiply:
(1/3^6) × (3^4/1)
This leaves us with 3^-2 inside the parentheses. When we apply the outer exponent of 3, we raise -2 by 3 to get -6.
Thus, the entire expression simplifies to 3^-6, which can be written as 1/3^6.
Since 9 is 3^2, 9^-6 is equivalent to (3^2)^-6, which simplifies to 3^-12. The negative power indicates the expression is in the denominator, so we have 3^6 in the numerator.
Therefore, the expression equivalent to the fraction is (3^6/9^15), and since we are looking for a negative value, the correct answer is option a) -(3^6/9^15).