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Answer :
To express the complex number [tex]\( 93 + \sqrt{-75} \)[/tex] in the correct notation, we'll follow these steps:
1. Simplify the square root of the negative number:
[tex]\(\sqrt{-75}\)[/tex] can be rewritten using the imaginary unit [tex]\(i\)[/tex], which is defined as [tex]\(\sqrt{-1}\)[/tex]. Therefore, [tex]\(\sqrt{-75} = \sqrt{75} \cdot i\)[/tex].
2. Simplify [tex]\(\sqrt{75}\)[/tex]:
We can simplify [tex]\(\sqrt{75}\)[/tex] by factoring it under the square root:
[tex]\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\][/tex]
3. Combine these results:
Substituting back, we have [tex]\(\sqrt{-75} = 5\sqrt{3} \cdot i\)[/tex].
4. Write the full expression for the complex number:
Substitute [tex]\(\sqrt{-75}\)[/tex] back into the expression to get:
[tex]\[
93 + \sqrt{-75} = 93 + 5\sqrt{3} \cdot i
\][/tex]
5. Identify the correct option:
Compare this expression to the given options:
- [tex]\(93 - 3i\sqrt{5}\)[/tex]
- [tex]\(93 + 5\sqrt{3}\)[/tex]
- [tex]\(93 + 5i\sqrt{3}\)[/tex]
- [tex]\(93 + 3i\sqrt{5}\)[/tex]
The expression [tex]\(93 + 5i\sqrt{3}\)[/tex] matches our result.
Therefore, the correct notation for the complex number [tex]\(93 + \sqrt{-75}\)[/tex] is [tex]\(\boxed{93 + 5i\sqrt{3}}\)[/tex].
1. Simplify the square root of the negative number:
[tex]\(\sqrt{-75}\)[/tex] can be rewritten using the imaginary unit [tex]\(i\)[/tex], which is defined as [tex]\(\sqrt{-1}\)[/tex]. Therefore, [tex]\(\sqrt{-75} = \sqrt{75} \cdot i\)[/tex].
2. Simplify [tex]\(\sqrt{75}\)[/tex]:
We can simplify [tex]\(\sqrt{75}\)[/tex] by factoring it under the square root:
[tex]\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\][/tex]
3. Combine these results:
Substituting back, we have [tex]\(\sqrt{-75} = 5\sqrt{3} \cdot i\)[/tex].
4. Write the full expression for the complex number:
Substitute [tex]\(\sqrt{-75}\)[/tex] back into the expression to get:
[tex]\[
93 + \sqrt{-75} = 93 + 5\sqrt{3} \cdot i
\][/tex]
5. Identify the correct option:
Compare this expression to the given options:
- [tex]\(93 - 3i\sqrt{5}\)[/tex]
- [tex]\(93 + 5\sqrt{3}\)[/tex]
- [tex]\(93 + 5i\sqrt{3}\)[/tex]
- [tex]\(93 + 3i\sqrt{5}\)[/tex]
The expression [tex]\(93 + 5i\sqrt{3}\)[/tex] matches our result.
Therefore, the correct notation for the complex number [tex]\(93 + \sqrt{-75}\)[/tex] is [tex]\(\boxed{93 + 5i\sqrt{3}}\)[/tex].
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