Answer :
To express the given complex number [tex]\(93 + \sqrt{-75}\)[/tex] in the correct notation, we'll follow these steps:
1. Identify the Imaginary Unit:
The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. So,
[tex]\[
\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1} = \sqrt{75} \cdot i
\][/tex]
2. Simplify [tex]\(\sqrt{75}\)[/tex]:
We need to simplify [tex]\(\sqrt{75}\)[/tex]. The number 75 can be broken down into its prime factors:
[tex]\[
75 = 3 \times 5 \times 5 = 3 \times 5^2
\][/tex]
Thus,
[tex]\[
\sqrt{75} = \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2} = \sqrt{3} \times 5 = 5\sqrt{3}
\][/tex]
3. Express the Complex Number:
Replacing [tex]\(\sqrt{-75}\)[/tex] in the original expression, we have:
[tex]\[
93 + \sqrt{-75} = 93 + 5\sqrt{3} \cdot i
\][/tex]
4. Select the Correct Notation:
The correct notation for the given complex number becomes:
[tex]\[
93 + 5i\sqrt{3}
\][/tex]
From the given options, the correct notation is [tex]\(93 + 5i\sqrt{3}\)[/tex].
1. Identify the Imaginary Unit:
The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. So,
[tex]\[
\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1} = \sqrt{75} \cdot i
\][/tex]
2. Simplify [tex]\(\sqrt{75}\)[/tex]:
We need to simplify [tex]\(\sqrt{75}\)[/tex]. The number 75 can be broken down into its prime factors:
[tex]\[
75 = 3 \times 5 \times 5 = 3 \times 5^2
\][/tex]
Thus,
[tex]\[
\sqrt{75} = \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2} = \sqrt{3} \times 5 = 5\sqrt{3}
\][/tex]
3. Express the Complex Number:
Replacing [tex]\(\sqrt{-75}\)[/tex] in the original expression, we have:
[tex]\[
93 + \sqrt{-75} = 93 + 5\sqrt{3} \cdot i
\][/tex]
4. Select the Correct Notation:
The correct notation for the given complex number becomes:
[tex]\[
93 + 5i\sqrt{3}
\][/tex]
From the given options, the correct notation is [tex]\(93 + 5i\sqrt{3}\)[/tex].
