https://file.aiquickdraw.com/mathbot/file/172403585190655r54mb1.jpg

Simplify the following expressions using the law of exponents

from PIL import Image
from pytesseract import pytesseract

# Load the uploaded image
image_path = '/mnt/data/172403585190655r54mb1.jpg'
img = Image.open(image_path)

# Use pytesseract to extract the math expression
text = pytesseract.image_to_string(img)

textIt appears that the image extraction did not yield the expected result. However, I can still simplify the expression manually by interpreting the mathematical notation visible in the image.

The given expression is:

\[
\frac{2x^4 y^{-4} z^{-3}}{3x^2 y^{-3} z^4}
\]

### Simplification Steps:

1. **Combine coefficients:**
   The coefficient of the numerator is \(2\), and the coefficient of the denominator is \(3\). Thus, the combined coefficient is:
   \[
   \frac{2}{3}
   \]

2. **Combine powers of \(x\):**
   Use the law of exponents: \( \frac{x^a}{x^b} = x^{a-b} \).
   \[
   x^{4-2} = x^2
   \]

3. **Combine powers of \(y\):**
   Similarly:
   \[
   y^{-4 - (-3)} = y^{-4 + 3} = y^{-1}
   \]

4. **Combine powers of \(z\):**
   Again, applying the exponent rule:
   \[
   z^{-3 - 4} = z^{-7}
   \]

### Final Simplified Expression:

\[
\frac{2}{3} x^2 y^{-1} z^{-7}
\]

This can also be written as:
\[
\frac{2x^2}{3yz^7}
\]

This is the simplified form of the given expression.

Would you like further clarification or have any questions?

---

Here are some related questions for further exploration:
1. How do the laws of exponents apply when dealing with negative exponents?
2. What is the process for simplifying expressions with multiple variables?
3. How do you handle zero exponents when simplifying?
4. What are the key differences in simplifying fractions versus whole number exponentiation?
5. How can fractional exponents be simplified in similar expressions?

**Tip:** Always apply exponent rules step by step to avoid errors, especially when combining terms with negative or fractional exponents.

Learn how to simplify exponential expressions using the laws of exponents with a detailed step-by-step solution. This problem covers key mathematical concepts including exponents, fractional exponents, and laws of exponents. Suitable for students in Grades 7-9.

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