Which of the following is an equivalent expression to (2^-3 x 9^3)^4 / 2^9 x 9^-10 with only positive exponents, generated by applying the Properties of Integer Exponents?
A. 2^3 / 9^2
B. 2^12 x 9^-12 / 2^9 x 9^-10
C. 2^-7 x 9^-1 / 2^9 x 9^-10
D. 2^3 x 9^2
To simplify the expression and get rid of negative exponents, we can use the following properties of integer exponents:
1. (a^m)^n = a^(m*n)
2. a^-m = 1/a^m
3. a^m / a^n = a^(m-n)
Let's simplify the given expression step by step:
(2^-3 x 9^3)^4 / 2^9 x 9^-10
Step 1:
Using property 1, simplify the expression inside the parentheses:
2^-3 x 9^3 = (2^-3)^4 x (9^3)^4
= 2^(-3*4) x 9^(3*4)
= 2^-12 x 9^12
Now the expression becomes:
(2^-12 x 9^12) / 2^9 x 9^-10
Step 2:
Using property 3, simplify the division:
(2^-12 x 9^12) / (2^9 x 9^-10) = 2^(-12-9) x 9^(12-(-10))
= 2^-21 x 9^22
Now the expression becomes:
2^-21 x 9^22
Step 3:
Using property 2, rewrite 2^-21 and 9^22:
2^-21 = 1/2^21
9^22 = (3^2)^22 = 3^(2*22) = 3^44
Now the expression becomes:
1/(2^21) x 3^44
This expression only has positive exponents. Therefore, the correct answer is option: D. 2^3 x 9^2.
To simplify the given expression and convert all exponents to positive, we can apply the properties of integer exponents.
Let's break down the given expression step by step:
Step 1: Simplify within parentheses
(2^-3 x 9^3)^4 = 2^(-3 x 4) x 9^(3 x 4) = 2^-12 x 9^12
Step 2: Simplify the denominator
2^9 x 9^-10
Step 3: Combine the numerator and denominator
(2^-12 x 9^12) / (2^9 x 9^-10)
Step 4: Apply the property of division with the same base
2^(-12 - 9) x 9^(12 - -10) = 2^-21 x 9^22
Hence, the equivalent expression with positive exponents is:
2^-21 x 9^22, which is equivalent to option C.
Therefore, the answer is C. 2^-7 x 9^-1 / 2^9 x 9^-10.