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Let's solve problem 9 from the image.

### Problem Statement:
Given that the absolute degree of \( M(x, y) = \sqrt{x^4 + y^4} \) is equal to 4, find the degree of \( P(x) = \sqrt{x^2 - 1} \).

### Step-by-Step Solution:

1. **Understanding the degree of a function**: 
   The degree of a polynomial or function refers to the highest power of the variable in the expression.

2. **Degree of \( M(x, y) \)**:
   - The expression \( M(x, y) = \sqrt{x^4 + y^4} \) is given.
   - Inside the square root, the highest degree term is \( x^4 \) or \( y^4 \). The degree of \( x^4 + y^4 \) is 4.
   - Since the degree of \( x^4 + y^4 \) is 4, taking the square root of this expression would halve the degree, making the expression \( M(x, y) \) have a degree of 2.

   However, the problem states that the absolute degree of \( M(x, y) \) is given as 4. This means that we must consider the expression inside the square root, not after taking the square root. 

   Thus, the degree of \( M(x, y) \) before taking the square root is indeed 4.

3. **Finding the degree of \( P(x) = \sqrt{x^2 - 1} \)**:
   - The expression inside the square root is \( x^2 - 1 \).
   - The degree of \( x^2 - 1 \) is 2.
   - Taking the square root would halve the degree. Thus, \( \sqrt{x^2 - 1} \) has a degree of \( \frac{2}{2} = 1 \).

### Final Answer:
The degree of \( P(x) = \sqrt{x^2 - 1} \) is **1**.

Would you like more details on this solution or have any questions?

### Related Questions:
1. What is the absolute degree of a polynomial in general?
2. How do we determine the degree of a multivariable polynomial?
3. What does it mean for a degree to be "absolute"?
4. How does the degree of a function change when applying different operations like addition or multiplication?
5. What is the impact on the degree when taking a square root of a polynomial?
6. Can we find the degree of a rational function?
7. How does the degree of a polynomial change when it is raised to a power?
8. What is the degree of a polynomial if one of its terms involves a square root?

**Tip**: Always double-check whether the degree mentioned is before or after an operation like taking a square root.

Explore how to determine the degree of the polynomial function P(x) = sqrt(x^2 - 1) through step-by-step explanation. This problem involves understanding polynomial degrees and the application of square root properties in mathematics.

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