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?
To solve this problem, we can simplify each power of a number separately and then multiply the results together.
First, let's simplify the expression (2^−3 ⋅ 9^3)^−4/2:
Since any number raised to the power of -1 is equal to 1 divided by that number raised to the positive exponent, we can rewrite 2^−3 as 1/2^3.
Similarly, we can rewrite 9^3 as 1/9^−3.
Thus, the expression becomes ((1/2^3) ⋅ (1/9^−3))^−4/2.
Using the product rule for exponents, we can rewrite (1/2^3) ⋅ (1/9^−3) as 1/(2^3 ⋅ 9^−3).
Then, applying the quotient rule for exponents, we can rewrite 9^−3 as 1/9^3.
So, the expression becomes 1/(2^3 ⋅ 1/9^3).
We can simplify this further by using the product rule for exponents again. 2^3 ⋅ (1/9^3) is equal to (2⋅1/9)^3.
Thus, the expression becomes 1/[(2/9)^3].
Finally, we can rewrite (2/9)^3 as (2^3)/(9^3), which simplifies the expression to 1/(8/729).
As dividing by a fraction is the same as multiplying by the reciprocal, we can rewrite 1/(8/729) as 1 * (729/8).
So, the expression simplifies to 729/8.
Now, let's simplify 9 ⋅ 9^−10:
Since any number raised to the power of 1 is equal to itself, we can rewrite 9 as 9^1.
Now, applying the product rule for exponents, we can combine the two terms with the same base (9) and add the exponents.
Thus, the expression simplifies to 9^11.
Finally, we need to multiply the results together.
729/8 * 9^11 is equal to (729/8) * (9^1 * 9^10).
Using the product rule for exponents, we can rewrite 9^1 * 9^10 as 9^11.
So, the expression simplifies to (729/8) * (9^11).
Multiplying a number by a fraction is the same as multiplying the number by the numerator and then dividing the result by the denominator, so the expression further simplifies to (729 * 9^11)/8.
Therefore, the equivalent expression, generated by applying the Properties of Integer Exponents, is (729 * 9^11)/8.