Question

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?

Answer 1

Thats incorrect try again

Answer 2

thats incorrect again, try again

Answer 3

To simplify the expression and apply the properties of integer exponents, let's break it down step by step.

First, let's simplify the base with negative exponents: 2^-3 is equal to 1/2^3, and 9^-10 is equal to 1/9^10.

Next, let's simplify the exponents within each parenthesis: (1/2^3 x 9^3)^-4 is equal to (1/8 x 729)^-4.

Now, let's simplify the second part of the expression: 2^9 is equal to 512.

Finally, let's rewrite the expression with only positive exponents: (1/8 x 729)^-4/512.

To simplify further, we need to apply the power of a quotient property: (a/b)^n = a^n / b^n.

Using this property, we can rewrite the expression as (1/8)^-4 x 729^-4/512.

Now, let's simplify each part separately: (1/8)^-4 is equal to 8^4 and 729^-4 is equal to (3^6)^-4.

Rewriting the expression again: 8^4 x (3^6)^-4 / 512.

Now, let's apply the property of negative exponents: (a^-n = 1/a^n).

Using this property, we can rewrite the expression as 8^4 x 1/(3^6)^4 / 512.

Simplifying further, 8^4 is equal to 4096, and (3^6)^4 is equal to 3^(6 x 4) or 3^24.

So the final equivalent expression with positive exponents is 4096 x 1/3^24 / 512.

Therefore, the answer is 4096 x 1/3^24 / 512.

Answer 4

Apologies for the previous incorrect response.

Let's break down the expression and simplify it step by step:

Starting with the base, we have 2^-3 * 9^3. Using the property (a^m)^n = a^(m*n), we can combine the exponents: (2^(-3*3)) * 9^3 = 2^-9 * 9^3.

Next, we have the expression raised to the power of -4. Using the property (a^m)^n = a^(m*n), we can multiply the exponents: (2^-9 * 9^3)^(-4) = 2^(-9*-4) * 9^(3*-4).

Simplifying further, we have 2^36 * 9^-12.

Now let's simplify the other part of the expression: 2^9 * 9^-10.

To combine these two expressions, we use the property (a^m * b^n) = (a*b)^(m+n): (2^36 * 9^-12 * 2^9 * 9^-10) = (2^36 * 2^9) * (9^-12 * 9^-10) = 2^(36+9) * 9^(-12-10) = 2^45 * 9^-22.

Now we need to express the expression with positive exponents. To do this, we use the property a^-n = 1/(a^n). Therefore, 9^-22 = 1/(9^22).

Putting it all together, the equivalent expression with positive exponents is 2^45 * 1/(9^22).

Answer 5

Apologies for the incorrect response again.

To simplify the expression (2^−3 X 9^3)^−4/2^9 X 9^−10 using the properties of integer exponents, let's break it down step by step:

First, let's simplify the base with negative exponents: 2^-3 is equal to 1/2^3, and 9^-10 is equal to 1/9^10.

Next, let's simplify the exponents within each parenthesis: (1/2^3 X 9^3)^-4 is equal to (1/8 X 729)^-4.

Now, let's simplify the second part of the expression: 2^9 is equal to 512.

Note: to handle negative exponents, we need to apply the reciprocal property: a^-n = 1/a^n.

Using this property, let's rewrite the expression: (8/729)^-4 / 512.

To simplify further, let's apply the power of a quotient property: (a/b)^n = a^n / b^n.

Using this property, we can rewrite the expression as (8/729)^-4 / 512^1.

Again, using the property of negative exponents, we can rewrite it as (729/8)^4 / 512.

Now, let's simplify each part separately: (729/8)^4 is equal to 729^4 / 8^4.

Finally, rewriting the expression: 729^4 / 8^4 / 512.

Therefore, an equivalent expression with only positive exponents is 729^4 / (8^4 * 512).