Apply the properties of integer exponents to generate an equivalent algebraic expression to this 3X with the small three on top and a -4 on top over 12 X with the small seven on top and why with the small four on top would be four small four on top and why with the small eight on top or would it be one over, 4X and a small 12 on top or one over 4X with a small four on top and a Y with a small eight on top or 3/12 X with a small four on top with a small eight on top

Answer 1

It seems that the expression you're asking about is quite complicated and may contain several components. I'll interpret it as best as I can based on what you've described. Let's denote the variables and their exponents in a more structured way.

Here's an interpretation of the expression:

1. \( \frac{3X^3}{12X^7Y^4} \)

Now, let's simplify this step by step using the properties of integer exponents:

Step 1: Simplifying the Coefficients

The coefficient part is \( \frac{3}{12} \): \[ \frac{3}{12} = \frac{1}{4} \]

Step 2: Simplifying the \( X \) Terms

Now, looking at the \( X \) terms: \[ \frac{X^3}{X^7} = X^{3-7} = X^{-4} \]

Step 3: Including the \( Y \) Term

Next, in the denominator, we have the \( Y^4 \): So the expression becomes: \[ \frac{1}{4} \cdot X^{-4} \cdot \frac{1}{Y^4} = \frac{1}{4Y^4} \cdot X^{-4} \]

Step 4: Writing in Positive Exponents

To express this with positive exponents, we rewrite \( X^{-4} \) as: \[ X^{-4} = \frac{1}{X^4} \]

Putting it all together, we have: \[ \frac{1}{4Y^4} \cdot \frac{1}{X^4} = \frac{1}{4X^4Y^4} \]

Thus, the equivalent simplified expression is: \[ \frac{1}{4X^4Y^4} \]

This is the final algebraic expression based on the provided input, applying the properties of integer exponents correctly. If this interpretation differs from what you intended, please clarify the components of your expression and I'll assist you further!