Which of the following is the equivalent expression to (2^-3x9^3)^-4/2^9x9^-10 that has been generated by applying the properties of integer exponents?
To simplify the given expression using the properties of integer exponents, we can apply the following rules:
1. For any nonzero base, a^0 = 1.
2. For any nonzero base, a^(-b) = 1/a^b.
3. a^b / a^c = a^(b-c).
Now, let's simplify the given expression step by step:
(2^-3 * 9^3)^(-4) / (2^9 * 9^-10)
First, apply the rule 2 to the numerator:
(1/(2^3 * 9^3))^(-4) / (2^9 * 9^-10)
Next, apply the rule 3 to the numerator:
1 / ((2^3 * 9^3)^4) / (2^9 * 9^-10)
Next, apply the rule 3 to the denominator:
1 / ((2^3 * 9^3)^4 / (2^9 * 9^-10))
Next, apply the rule 2 to the denominator:
1 / ((2^3 * 9^3)^4 / (2^9 * 1 / 9^10))
Now, let's simplify the expression within the parentheses:
1 / (((2^3)^4 * (9^3)^4) / (2^9 * 1 / 9^10))
Apply the power rule to the bases within parentheses:
1 / ((2^12 * 9^12) / (2^9 * 1 / 9^10))
Simplify the expression further:
1 / ((2^12 * 9^12) / (2^9 * 9^10))
Since the denominator is still a fraction, we can rewrite it as follows:
1 / (2^12 * 9^12) * (2^9 * 9^10)
Now, we can combine the exponents using the rule 3:
1 / (2^(12 - 9) * 9^(12 + 10))
Simplify the exponents:
1 / (2^3 * 9^22)
Finally, rewrite the expression using the power rule:
1 / 8 * 9^(-22)
Thus, the equivalent expression that has been generated by applying the properties of integer exponents is 9^(-22) / 8.