Certainly! Let's go through the process of rewriting and simplifying the expression [tex]\(\sqrt{-75}\)[/tex] using imaginary numbers.
1. Understanding the Imaginary Unit:
The imaginary unit [tex]\(i\)[/tex] is defined by [tex]\(i = \sqrt{-1}\)[/tex]. This means that for any positive number [tex]\(a\)[/tex], [tex]\(\sqrt{-a} = \sqrt{a} \cdot i\)[/tex].
2. Rewrite the Given Expression:
We start with the given expression [tex]\(\sqrt{-75}\)[/tex].
3. Express [tex]\(\sqrt{-75}\)[/tex] Using [tex]\(i\)[/tex]:
Recognize that we can separate the negative sign from the number as follows:
[tex]\[
\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1}
\][/tex]
Since [tex]\(\sqrt{-1} = i\)[/tex], we can rewrite this as:
[tex]\[
\sqrt{-75} = \sqrt{75} \cdot i
\][/tex]
4. Simplify the Radical:
Now, let's simplify [tex]\(\sqrt{75}\)[/tex].
[tex]\[
75 = 25 \cdot 3
\][/tex]
Therefore, it follows that:
[tex]\[
\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3}= 5 \cdot \sqrt{3}
\][/tex]
So we can substitute this back into our expression:
[tex]\[
\sqrt{-75} = 5 \cdot \sqrt{3} \cdot i
\][/tex]
5. Determining the Value of [tex]\(\sqrt{3}\)[/tex]:
The value of [tex]\(\sqrt{3}\)[/tex] approximately equals [tex]\(1.732\)[/tex].
6. Calculate the Simplified Form:
Using this approximation, we can calculate:
[tex]\[
5 \cdot \sqrt{3} \approx 5 \cdot 1.732 = 8.66
\][/tex]
So, the simplified form of [tex]\(\sqrt{-75}\)[/tex] is approximately:
[tex]\[
8.66i
\][/tex]
Thus, the expression [tex]\(\sqrt{-75}\)[/tex] can be rewritten as a complex number in its simplified form:
[tex]\[
8.66i
\][/tex]
