To determine which radical expression correctly represents the complex number [tex]\(7 + 9i\)[/tex], we need to analyze the given options and understand how imaginary numbers work. Recall that [tex]\(i\)[/tex] is defined as the square root of [tex]\(-1\)[/tex], i.e., [tex]\(i = \sqrt{-1}\)[/tex].
Let's explore each option step by step:
1. Option A: [tex]\(7 + \sqrt{-3}\)[/tex]
- Here, [tex]\(\sqrt{-3}\)[/tex] can be written as [tex]\(\sqrt{3} \cdot i\)[/tex].
- Thus, the expression becomes [tex]\(7 + \sqrt{3} \cdot i\)[/tex].
- This does not match [tex]\(7 + 9i\)[/tex] since [tex]\(\sqrt{3} \)[/tex] is not equal to [tex]\(9\)[/tex].
2. Option B: [tex]\(7 - \sqrt{-9}\)[/tex]
- Here, [tex]\(\sqrt{-9}\)[/tex] can be written as [tex]\(\sqrt{9} \cdot i\)[/tex], which equals [tex]\(3i\)[/tex].
- So this expression becomes [tex]\(7 - 3i\)[/tex].
- This does not match [tex]\(7 + 9i\)[/tex] as it involves subtraction and different imaginary parts.
3. Option C: [tex]\(7 + \sqrt{-9}\)[/tex]
- Here, [tex]\(\sqrt{-9}\)[/tex] can be written as [tex]\(\sqrt{9} \cdot i\)[/tex], which equals [tex]\(3i\)[/tex].
- So this expression becomes [tex]\(7 + 3i\)[/tex].
- This does not match [tex]\(7 + 9i\)[/tex] since the imaginary part is [tex]\(3\)[/tex] instead of [tex]\(9\)[/tex].
4. Option D: [tex]\(7 + \sqrt{-81}\)[/tex]
- Here, [tex]\(\sqrt{-81}\)[/tex] can be written as [tex]\(\sqrt{81} \cdot i\)[/tex], which equals [tex]\(9i\)[/tex].
- So this expression becomes [tex]\(7 + 9i\)[/tex].
- This exactly matches our given complex number [tex]\(7 + 9i\)[/tex].
Therefore, the correct radical expression that represents the complex number [tex]\(7 + 9i\)[/tex] is:
D. [tex]\(7 + \sqrt{-81}\)[/tex]
