if 9^y=x, which of the following represents 9^y-9^(y-3)
A) 1/x^3
B) x^3
C) 729x
D) 729/728 x
E) 728/729 x
9^y = x, so
9^y-9^(y-3) = 9^y - 9^y/9^3 = x - x/729 = ...
To solve this problem, we can start by simplifying the expression 9^y - 9^(y-3).
First, let's recall the properties of exponents. When we subtract two exponential expressions with the same base, we can divide the exponents. This can be represented as:
a^b / a^c = a^(b-c)
Using this property, we can simplify the expression as follows:
9^y - 9^(y-3) = 9^y / 9^(y-3)
Now, let's rewrite 9^(y-3) as 9^y / 9^3. We divide by 9^3 because dividing by 9^(y-3) is the same as multiplying by its reciprocal:
9^y - 9^(y-3) = 9^y / (9^y / 9^3)
Simplifying further, we can rewrite 9^y as (9^3)^y:
9^y - 9^(y-3) = (9^3)^y / (9^y / 9^3)
Now, let's simplify the expression with exponents:
9^y - 9^(y-3) = 729^y / (9^y / 9^3)
Simplifying the expression inside the parentheses by dividing 9^y by 9^y:
9^y - 9^(y-3) = 729^y * (9^3 / 9^y)
Next, simplify the expression inside the parentheses by subtracting the exponents:
9^y - 9^(y-3) = 729^y * 9^(3-y)
Finally, simplify further using the property of multiplying two exponential expressions with the same base:
9^y - 9^(y-3) = 729^y+1 * 9^(3-y)
Now, we can compare this result to the answer choices:
A) 1/x^3
B) x^3
C) 729x
D) 729/728x
E) 728/729x
None of the answer choices match the simplified expression we derived. Therefore, none of the given answer choices represent 9^y - 9^(y-3).