Question: Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents..

To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:

1. Any non-zero number raised to the power of 0 is equal to 1: a^0 = 1, where a ≠ 0.

2. When dividing two numbers with the same base raised to different exponents, we subtract the exponents: a^m / a^n = a^(m - n), where a ≠ 0.

3. When raising a number with a negative exponent to the power of a positive exponent, we can move the base to the denominator and change the sign of the exponent: a^(-m) = 1/a^m, where a ≠ 0.

Let's apply these properties step by step:

First, we have 4^0 ⋅ 5^(-2)/(2^(-3)).

According to property 1, 4^0 = 1.

Next, we can apply property 3 to convert 5^(-2) to 1/5^2 = 1/25.

Similarly, we can apply property 3 to convert (2^(-3)) to 1/(2^3) = 1/8.

Now, our expression becomes 1 ⋅ 1/25 / 1/8.

Applying property 2, we subtract the exponents in the denominator when dividing: 1 ⋅ 1/25 * 8/1 = 8/25.

Finally, the simplified fraction with no remaining exponents is 8/25.

that was incorrect, try again

Apologies for the incorrect response. Let's try again.

To generate equivalent expressions with positive exponents, we can use the properties of integer exponents:

1. Any non-zero number raised to the power of 0 is equal to 1: a^0 = 1, where a ≠ 0.

2. When dividing two numbers with the same base raised to different exponents, we subtract the exponents: a^m / a^n = a^(m - n), where a ≠ 0.

3. When raising a number with a negative exponent to the power of a positive exponent, we can move the base to the denominator and change the sign of the exponent: a^(-m) = 1/a^m, where a ≠ 0.

Let's apply these properties step by step:

The expression is 4^0 ⋅ 5^(-2)/(−2)^(-3).

According to property 1, 4^0 = 1.

Applying property 3, we can convert 5^(-2) to 1/5^2 = 1/25.

Similarly, we can convert (−2)^(-3) to 1/(−2)^3 = 1/(-8) = -1/8.

Now, the expression becomes 1 ⋅ 1/25 / (-1/8).

We can simplify this further. When dividing by a fraction, we multiply by its reciprocal. So, the expression becomes 1 ⋅ 1/25 * (-8/1) = -8/25.

Therefore, the simplified fraction with no remaining exponents is -8/25.

that was correct, thank you

You're welcome! I'm glad the second response was correct. If you have any more questions, feel free to ask.