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The expression you provided is:

\[ x^{\frac{1}{2}}(y + 5) + \sqrt{-z} \]

Given values:
- \( x = \frac{1}{2} \)
- \( y = -9 \)
- \( z = 125 \)

To find the value of this expression, let's evaluate each part step by step:

1. **Square Root of \(x\)**: \( \sqrt{x} = \sqrt{\frac{1}{2}} \)
2. **Evaluate \(y + 5\)**: \( y + 5 = -9 + 5 = -4 \)
3. **Product of \( \sqrt{x} \) and \( (y + 5) \)**: \( \sqrt{\frac{1}{2}} \cdot (-4) \)
4. **Square Root of \(-z\)**: Since \( z = 125 \), \( \sqrt{-125} \) is not defined in the set of real numbers. It would involve imaginary numbers.

Let's calculate the real number components and clarify the nature of \( \sqrt{-z} \) using Python.The calculation for \( \sqrt{-125} \) caused a math domain error because it's not defined within the real numbers. To correctly handle this, we should treat it as an imaginary number. Let's compute the expression by taking the imaginary unit \( i \) into account, where \( i^2 = -1 \):

\[ \sqrt{-125} = \sqrt{125}i = 5\sqrt{5}i \]

Now, combining this with the rest of the expression, we'll calculate the correct values.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to know more about complex numbers or have any other questions on a different topic? Here are some questions you might consider:

1. What are imaginary numbers and how are they used in calculations?
2. How can you find the square root of a negative number?
3. What is the significance of the imaginary unit \(i\) in complex numbers?
4. Could you explain how to simplify expressions involving square roots of fractions?
5. How do operations with complex numbers differ from those with real numbers?

**Tip:** When dealing with square roots of negative numbers, remember to include the imaginary unit \(i\), where \(i^2 = -1\), to ensure the expression remains valid within the set of complex numbers.

What is the value of the expression x^(1/2)(y + 5) + √(-z) when x = 1/2, y = -9, and z = 125?

Learn how to evaluate the expression x^(1/2)(y + 5) + √(-z) for values x = 1/2, y = -9, and z = 125. This problem explores key algebraic concepts like radicals and complex numbers, and includes the imaginary unit. Suitable for students in Grades 10-12.

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