Question

simplify to standard form
\sqrt{-10000}
options: 100i, 10i, -10i, -100i

Explanation:

Step1: Recall the imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). So we can rewrite \( \sqrt{-100} \) as \( \sqrt{100 \times (-1)} \).

Step2: Use the property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0, b\geq0 \), and here we extend it for complex numbers), we have \( \sqrt{100\times(-1)}=\sqrt{100}\times\sqrt{-1} \).

Step3: Simplify the square roots

We know that \( \sqrt{100} = 10 \) and \( \sqrt{-1}=i \), so \( \sqrt{100}\times\sqrt{-1}=10i \). Wait, but looking at the options, maybe there is a typo and the original problem is \( \sqrt{-10000} \)? Let's check. If it's \( \sqrt{-10000} \), then \( \sqrt{10000\times(-1)}=\sqrt{10000}\times\sqrt{-1}=100i \). Wait, the options have 100i. Let's re - evaluate.

If the problem is \( \sqrt{-100} \), \( \sqrt{-100}=\sqrt{100}\times\sqrt{-1} = 10i \), but the options have 100i. Maybe the problem is \( \sqrt{-10000} \). \( \sqrt{-10000}=\sqrt{10000}\times\sqrt{-1}=100i \). So the correct option should be the one with 100i.

Answer:

B. 100i (assuming the problem might have a typo and the radicand is - 10000 instead of - 100, or if the original problem's radicand is - 10000, then the answer is 100i. If we strictly take the given problem as \( \sqrt{-100} \), there is a mismatch, but based on the options, 100i is the most probable. So the answer is the option with 100i, for example, if the options are: A. 1000i, B. 100i, C. - 100i, D. - 1000i, then the answer is B. 100i)