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Answer :
Sure, let's work through this step by step.
We need to simplify the expression [tex]\(\sqrt{-48} - 89\)[/tex].
1. Simplify [tex]\(\sqrt{-48}\)[/tex]:
The square root of a negative number involves imaginary numbers. The imaginary unit is denoted as [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
- First, separate the negative sign: [tex]\(\sqrt{-48} = \sqrt{-1 \times 48}\)[/tex].
- This can be rewritten using the properties of square roots: [tex]\(\sqrt{-48} = \sqrt{-1} \times \sqrt{48}\)[/tex].
- [tex]\(\sqrt{-1}\)[/tex] is [tex]\(i\)[/tex], so we have: [tex]\(\sqrt{-48} = i \times \sqrt{48}\)[/tex].
2. Simplify [tex]\(\sqrt{48}\)[/tex]:
We can factor 48 to simplify: [tex]\(48 = 16 \times 3\)[/tex].
- Therefore, [tex]\(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}\)[/tex].
- [tex]\(\sqrt{16} = 4\)[/tex], so: [tex]\(\sqrt{48} = 4 \times \sqrt{3}\)[/tex].
3. Combine the results:
Substitute back to get [tex]\(\sqrt{-48} = i \times 4 \times \sqrt{3} = 4i \times \sqrt{3}\)[/tex].
4. Express the original expression:
Now, substitute [tex]\(\sqrt{-48}\)[/tex] back into the expression [tex]\(\sqrt{-48} - 89\)[/tex]:
[tex]\(-89 + 4i \times \sqrt{3}\)[/tex].
5. Conclusion:
The correct notation for the expression is [tex]\(-89 + 4i \sqrt{3}\)[/tex].
From the given options, this matches with:
[tex]\(-89 + 4i \sqrt{3}\)[/tex].
We need to simplify the expression [tex]\(\sqrt{-48} - 89\)[/tex].
1. Simplify [tex]\(\sqrt{-48}\)[/tex]:
The square root of a negative number involves imaginary numbers. The imaginary unit is denoted as [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
- First, separate the negative sign: [tex]\(\sqrt{-48} = \sqrt{-1 \times 48}\)[/tex].
- This can be rewritten using the properties of square roots: [tex]\(\sqrt{-48} = \sqrt{-1} \times \sqrt{48}\)[/tex].
- [tex]\(\sqrt{-1}\)[/tex] is [tex]\(i\)[/tex], so we have: [tex]\(\sqrt{-48} = i \times \sqrt{48}\)[/tex].
2. Simplify [tex]\(\sqrt{48}\)[/tex]:
We can factor 48 to simplify: [tex]\(48 = 16 \times 3\)[/tex].
- Therefore, [tex]\(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}\)[/tex].
- [tex]\(\sqrt{16} = 4\)[/tex], so: [tex]\(\sqrt{48} = 4 \times \sqrt{3}\)[/tex].
3. Combine the results:
Substitute back to get [tex]\(\sqrt{-48} = i \times 4 \times \sqrt{3} = 4i \times \sqrt{3}\)[/tex].
4. Express the original expression:
Now, substitute [tex]\(\sqrt{-48}\)[/tex] back into the expression [tex]\(\sqrt{-48} - 89\)[/tex]:
[tex]\(-89 + 4i \times \sqrt{3}\)[/tex].
5. Conclusion:
The correct notation for the expression is [tex]\(-89 + 4i \sqrt{3}\)[/tex].
From the given options, this matches with:
[tex]\(-89 + 4i \sqrt{3}\)[/tex].
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