Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

To simplify the expression \( \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \) using the properties of integer exponents, we can start by applying the rules of exponents to each term.

1. First, evaluate \( 4^0 \): \[ 4^0 = 1 \] This simplifies the expression to \( \frac{1 \cdot 5^{-2}}{(-2)^{-3}} = \frac{5^{-2}}{(-2)^{-3}} \).

2. Rewrite \( 5^{-2} \) using the property \( a^{-n} = \frac{1}{a^n} \): \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

3. Rewrite \( (-2)^{-3} \) similarly: \[ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} \]

4. Substitute these back into the fraction: \[ \frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{25}}{\frac{1}{-8}} = \frac{1}{25} \cdot \left(-8\right) \]

5. This simplifies to: \[ \frac{-8}{25} \]

Thus, the simplified expression is: \[ \frac{-8}{25} \]

The final answer is: \[ \frac{-8}{25} \]

Answer 2

no

Answer 3

Let's go through the expression again step by step, ensuring everything is correct and simplifying with only positive exponents.

We start with the original expression:

\[ \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \]

Step 1: Simplify Each Component

1. Evaluate \( 4^0 \): \[ 4^0 = 1 \]

   Therefore, the expression simplifies to:

   \[ \frac{1 \cdot 5^{-2}}{(-2)^{-3}} = \frac{5^{-2}}{(-2)^{-3}} \]

2. Rewrite \( 5^{-2} \) using the property \( a^{-n} = \frac{1}{a^n} \): \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

3. Rewrite \( (-2)^{-3} \) similarly: \[ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} \]

Now we can substitute these into the expression:

Step 2: Substitute Back

Now we have:

\[ \frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{25}}{\frac{1}{-8}} \]

This is equal to:

\[ \frac{1}{25} \cdot (-8) = \frac{-8}{25} \]

Final Result

The final result is:

\[ \frac{-8}{25} \]

This is a simplified fraction with no remaining exponents, and it is the correct solution.

If you meant something different, please specify, and I would be happy to assist further!