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 Integer Exponents? (1 point) (2 ^ 12 * 9 ^ - 12)/(2 ^ 9 * 9 ^ - 10); (2 ^ - 7 * 9 ^ - 1)/(2 ^ 9 * 9 ^ - 10); 2 ^ 3 * 9 ^ 2 C (2 ^ 3)/(9 ^ 2)

Question

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 Integer Exponents? (1 point) (2 ^ 12 * 9 ^ - 12)/(2 ^ 9 * 9 ^ - 10); (2 ^ - 7 * 9 ^ - 1)/(2 ^ 9 * 9 ^ - 10); 2 ^ 3 * 9 ^ 2 C (2 ^ 3)/(9 ^ 2)

Answer 1

The equivalent expression with only positive exponents, generated by applying the Properties of Integer Exponents, is (2 ^ 12 * 9 ^ -12) / (2 ^ 9 * 9 ^ -10).

Answer 2

To simplify the given expression and generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents.

Starting with the given expression: ((2 ^ - 3 * 9 ^ 3) ^ - 4)/(2 ^ 9 * 9 ^ - 10)

Step 1: Deal with the negative exponents by applying the rule: a ^ -n = 1 / (a ^ n)
Rewriting the expression: ((1 / 2 ^ 3 * 1 / 9 ^ 3) ^ - 4)/(2 ^ 9 * 1 / 9 ^ 10)

Step 2: Simplify the negative exponent by applying the rule: (a / b) ^ -n = (b / a) ^ n
Rewriting the expression: ((1 / 2 ^ 3 * 1 / 9 ^ 3) ^ 4)/(2 ^ 9 * 9 ^ 10)

Step 3: Simplify the exponents by applying the rule: (a ^ m * b ^ n) / (a ^ p * b ^ q) = a ^ (m - p) * b ^ (n - q)
Rewriting the expression: (1 / (2 ^ (3 * 4) * 9 ^ (3 * 4)))/(2 ^ (9 - 0) * 9 ^ (10 - 3))

Simplifying further: (1 / (2 ^ 12 * 9 ^ 12))/(2 ^ 9 * 9 ^ 7)

Thus, the equivalent expression with only positive exponents is:
(2 ^ 12 * 9 ^ -12)/(2 ^ 9 * 9 ^ -7)

Therefore, the correct answer is: (2 ^ 12 * 9 ^ -12)/(2 ^ 9 * 9 ^ -7)

Answer 3

To simplify the given expression using the Properties of Integer Exponents, we can apply the following rules:

1. (a^m)^n = a^(m*n)
2. a^m * a^n = a^(m+n)
3. a^m / a^n = a^(m-n)

Let's break down the given expression step by step:

((2^-3 * 9^3)^-4) / (2^9 * 9^-10)

First, let's apply the exponent rules to simplify the numerator of the expression:

(2^-3 * 9^3)^-4

Using rule 1, we have:

= 2^(-3*-4) * 9^(3*-4)

Simplifying the exponents:

= 2^12 * 9^-12

Now, let's simplify the denominator of the expression:

2^9 * 9^-10

Using rule 2, we add the exponents:

= 2^(9+0) * 9^(-10)

Simplifying:

= 2^9 * 9^-10

Finally, putting the simplified numerator and denominator together, we get:

(2^12 * 9^-12) / (2^9 * 9^-10)

Simplifying further, we can use rule 3 to divide the exponents:

= 2^(12-9) * 9^(-12-(-10))

= 2^3 * 9^(-2)

Therefore, the equivalent expression with only positive exponents is:

2^3 * 9^2