To express [tex]\(\sqrt{-225}\)[/tex] in its simplest terms, we need to consider the properties of square roots and imaginary numbers.
1. Understand the square root of a negative number: The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex]. Therefore, for any negative number [tex]\(-a\)[/tex], [tex]\(\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1}\)[/tex].
2. Break down [tex]\(\sqrt{-225}\)[/tex]:
- Start by noting that [tex]\(-225\)[/tex] can be written as [tex]\(225 \times -1\)[/tex].
- Thus, [tex]\(\sqrt{-225} = \sqrt{225 \times -1}\)[/tex].
3. Separate the product inside the square root:
- This can be split into the product of two square roots: [tex]\(\sqrt{225} \)[/tex] and [tex]\(\sqrt{-1}\)[/tex].
- So, [tex]\(\sqrt{225 \times -1} = \sqrt{225} \times \sqrt{-1}\)[/tex].
4. Evaluate each square root individually:
- First, find [tex]\(\sqrt{225}\)[/tex]. Since [tex]\(225 = 15^2\)[/tex], we have [tex]\(\sqrt{225} = 15\)[/tex].
- Next, recall that [tex]\(\sqrt{-1} = i\)[/tex].
5. Combine the results:
- Multiplying these results together gives [tex]\(\sqrt{-225} = 15 \times i\)[/tex].
Thus, the simplest term for [tex]\(\sqrt{-225}\)[/tex] is [tex]\(\boxed{15i}\)[/tex].
