Which of the following is an equivalent expression to ((2 ^ - 3 * 9 ^ 3) ^ - 4)/(2 ^ 9 * 9 ^ - 10) with only positive exponents, generated by applying the Properties of Integer Exponents? (1 point)
(2 ^ 12 * 9 ^ - 12)/(2 ^ 9 * 9 ^ - 10)
(2 ^ - 7 * 9 ^ - 1)/(2 ^ 9 * 9 ^ - 10)
2 ^ 3 * 9 ^ 2
(2 ^ 3)/(9^ 2)
To simplify the given expression using the properties of integer exponents, we can apply the following rules:
1. Product of powers with the same base: a^m * a^n = a^(m+n)
2. Quotient of powers with the same base: a^m / a^n = a^(m-n)
3. Power of a power: (a^m)^n = a^(m*n)
Now let's simplify the given expression step by step:
((2^-3 * 9^3)^-4) / (2^9 * 9^-10)
Using the power of a power rule, we can simplify the inner parenthesis:
(2^(-3*-4) * 9^(3*-4)) / (2^9 * 9^-10)
Simplifying the exponents in the parenthesis:
(2^12 * 9^-12) / (2^9 * 9^-10)
Now let's apply the quotient of powers rule to simplify the expression further:
2^(12-9) * 9^(-12-(-10))
2^3 * 9^(-12+10)
Simplifying the exponents:
2^3 * 9^-2
Therefore, the equivalent expression with only positive exponents is:
(2^3)/(9^2)