https://r2.erweima.ai/mathbot/image/1723926221115ktbcdq3v.jpg

Let's analyze the expression from the image step by step to determine if the result is \( 2950 \sqrt{5} \).

The expression is:

\[
-250 \cdot (-3.5) + 1000 \div \left(-5^{\frac{1}{2}} - \sqrt{25}\right) - 500 \cdot 5^{\frac{1}{2}} \cdot \sqrt{100}
\]

Let's break it down and simplify it:

1. **First part:** \( -250 \cdot (-3.5) \)
   \[
   -250 \times -3.5 = 875
   \]

2. **Second part:** \( 1000 \div \left(-5^{\frac{1}{2}} - \sqrt{25}\right) \)
   - \( 5^{\frac{1}{2}} \) is the square root of 5: \( \sqrt{5} \)
   - \( \sqrt{25} = 5 \)
   - So the expression inside the parenthesis becomes \( -\sqrt{5} - 5 \).
   - The expression is now \( 1000 \div \left(-\sqrt{5} - 5\right) \).

3. **Third part:** \( -500 \cdot 5^{\frac{1}{2}} \cdot \sqrt{100} \)
   - \( 5^{\frac{1}{2}} = \sqrt{5} \)
   - \( \sqrt{100} = 10 \)
   - So the expression becomes \( -500 \times \sqrt{5} \times 10 = -5000 \times \sqrt{5} \).

Putting everything together:

\[
875 + \frac{1000}{-\sqrt{5} - 5} - 5000\sqrt{5}
\]

Now, the second term \( \frac{1000}{-\sqrt{5} - 5} \) needs to be simplified further, but it’s clear that this term won't cancel out the \( -5000\sqrt{5} \) term. Therefore, the resulting expression cannot simply be \( 2950\sqrt{5} \). Instead, it should result in a more complex form. Thus, the result of the given expression is not \( 2950\sqrt{5} \).

Would you like to see the detailed steps of the calculation or explore any part of the expression further? Here are some related questions to expand on this:

1. How would you simplify \( \frac{1000}{-\sqrt{5} - 5} \)?
2. Can the expression be factorized or further simplified to match the given form \( 2950\sqrt{5} \)?
3. How do square roots behave under multiplication or division?
4. What techniques are used to rationalize a denominator?
5. How does changing a term in the expression affect the final result?

**Tip:** When working with roots and complex fractions, always consider rationalizing the denominator to simplify the expression.

Explore the detailed analysis of a complex mathematical expression involving arithmetic operations, square roots, and complex fractions. Understand step-by-step how the expression is broken down and simplified. Suitable for advanced high school students studying algebra and advanced arithmetic.

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