Wich of the following equivalent expression to (2^-3×9^3)^-4/2^9×9^-10 with only positive exponents generated by applying the properties of integer exponents
2^-7×9^-1/2^9×9^-10
2^3/9^2
2^3×9^2
2^12×9^-12/2^9×9-10
To simplify the given expression and apply the properties of integer exponents, we can work step by step:
(2^-3 × 9^3)^-4 / 2^9 × 9^-10
First, let's simplify the expression inside the parentheses:
2^-3 × 9^3 = (1/2^3) × (9^3) = 1/8 × 729 = 729/8
Now, substitute this value back into the original expression and simplify further:
(729/8)^-4 / 2^9 × 9^-10
To simplify (729/8)^-4, we need to apply the negative exponent to the numerator and denominator:
(8/729)^4 / 2^9 × 9^-10
Now, we can simplify the expression by combining the like terms in the numerator and denominator:
(8^4 / 729^4) / 2^9 × 9^-10
Next, let's simplify the numerator and denominator separately:
8^4 = 8 × 8 × 8 × 8 = 4096
729^4 = 729 × 729 × 729 × 729 = 2028240951369
Substituting these values back into the expression:
(4096 / 2028240951369) / 2^9 × 9^-10
To simplify further, let's simplify 9^-10:
9^-10 = 1/9^10 = 1/3486784401
Substituting this value back into the expression:
(4096 / 2028240951369) / 2^9 × (1/3486784401)
Now, we can simplify the expression by combining the terms:
(4096 / 2028240951369) / 2^9 × (1/3486784401)
= 0.000002014522 × (1/512) × (1/3486784401)
= 0.000000003936 × (1/3486784401)
= 0.000000003936 / 3486784401
Therefore, the equivalent expression with only positive exponents is:
2^-7 × 9^-1 / 2^9 × 9^-10