Simplify $-\sqrt{-625}$: A Step-by-Step Solution

Let's tackle the problem of simplifying the expression $-\sqrt{-625}$. This involves understanding imaginary numbers and how to manipulate them. This detailed walkthrough will help clarify each step, ensuring you grasp the underlying concepts.

Understanding Imaginary Numbers

Before we dive into the simplification, let's briefly discuss imaginary numbers. The imaginary unit, denoted as i, is defined as the square root of -1. Mathematically, this is expressed as $i = \sqrt{-1}$. This concept is crucial because it allows us to deal with the square roots of negative numbers, which are not defined within the realm of real numbers. When we encounter a negative number inside a square root, we factor out -1 and replace $\sqrt{-1}$ with i. For example, $\sqrt{-9}$ can be written as $\sqrt{9 \cdot -1} = \sqrt{9} \cdot \sqrt{-1} = 3i$. This basic understanding is fundamental to solving the problem at hand and many other problems involving complex numbers. Knowing how to properly handle imaginary units will allow you to simplify more complex expressions with confidence and accuracy. Remember, the key is to always factor out the -1 and replace it with i, thereby transforming the expression into a form that is easier to work with.

Step-by-Step Simplification of $-\sqrt{-625}$

Let's simplify the expression $-\sqrt{-625}$ step-by-step. This will involve breaking down the problem into smaller, manageable parts to ensure clarity.

1. Factor out -1: The first step is to recognize that we have a negative number inside the square root. We can rewrite the expression as follows:

   $-\sqrt{-625} = -\sqrt{625 \cdot -1}$

   This separates the negative sign from the positive number, allowing us to deal with the imaginary unit.

2. Apply the square root property: Now, we can use the property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ to separate the square root:

   $-\sqrt{625 \cdot -1} = -(\sqrt{625} \cdot \sqrt{-1})$

   This step isolates the square root of -1, which is the imaginary unit i.

3. Evaluate the square root of 625: We know that $25 \cdot 25 = 625$, so $\sqrt{625} = 25$. Thus, we can replace $\sqrt{625}$ with 25:

   $-(25 \cdot \sqrt{-1})$

4. Replace $\sqrt{-1}$ with i: By definition, $\sqrt{-1} = i$. So, we substitute i into the expression:

   $-(25 \cdot i) = -25i$

   Therefore, the simplified form of $-\sqrt{-625}$ is $-25i$.

Detailed Explanation of Each Step

Let's dive deeper into each step to ensure full comprehension. The initial expression is $-\sqrt{-625}$. Our primary goal is to express this in its simplest form, which involves handling the negative sign inside the square root. By factoring out -1, we transform the expression into a product that includes $\sqrt{-1}$, which is defined as the imaginary unit i. Recognizing this is crucial. Then we apply the property of square roots to separate the factors, making it easier to manage each component individually. Evaluating $\sqrt{625}$ requires finding a number that, when multiplied by itself, equals 625. Since $25 \cdot 25 = 625$, the square root of 625 is 25. Substituting this value back into the expression, we are left with $-(25 \cdot \sqrt{-1})$. The final step involves replacing $\sqrt{-1}$ with i. Thus, the expression becomes $-(25i)$, which simplifies to $-25i$. This step-by-step breakdown clarifies how we move from the initial complex-looking expression to its simplest imaginary form. The key is to methodically apply the properties of square roots and imaginary numbers to break down the problem into smaller, more manageable parts. This ensures accuracy and understanding.

Why Other Options Are Incorrect

To further solidify our understanding, let's examine why the other provided options are incorrect:

• A. $8i\sqrt{5}$: This option is incorrect because it seems to involve some misunderstanding of how to factor the number inside the square root. There's no valid mathematical operation that would lead to this result starting from $-\sqrt{-625}$. The number 625 is a perfect square, and there's no need to introduce any additional square roots.
• C. $25i$: This is the positive version of the correct answer. However, the original expression has a negative sign outside the square root, which must be preserved throughout the simplification process. Neglecting this negative sign leads to an incorrect result.
• D. $-8i\sqrt{5}$: Similar to option A, this option is incorrect due to an improper factoring or simplification process. It incorrectly introduces a square root of 5 and does not account for the correct square root of 625. There's no logical mathematical pathway from the original expression to this result.

Understanding why these options are wrong reinforces the correct methodology and helps prevent similar mistakes in future problems. The critical point is to meticulously follow the correct algebraic and arithmetic steps, ensuring that each transformation is valid and justified. Recognizing common errors is as important as knowing the correct solution.

Conclusion

Therefore, the correct answer is B. $-25i$. This detailed explanation breaks down each step involved in simplifying the expression $-\sqrt{-625}$, clarifying the role of imaginary numbers and how to manipulate them correctly. By understanding each step and why other options are incorrect, you can confidently tackle similar problems in the future.

For further reading and to deepen your understanding of imaginary numbers, visit Khan Academy's Complex Numbers.