To simplify the expression [tex]\(6 + \sqrt{-80}\)[/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]. Therefore,
[tex]\[
\sqrt{-80} = \sqrt{80} \cdot \sqrt{-1} = \sqrt{80} \cdot i
\][/tex]
2. Simplify the radical [tex]\( \sqrt{80} \)[/tex]: We can factor 80 into
[tex]\(16 \cdot 5\)[/tex], and then use the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Thus,
[tex]\[
\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \cdot \sqrt{5}
\][/tex]
3. Combine the imaginary part: Now, substituting this back into the expression,
[tex]\[
\sqrt{-80} = 4 \sqrt{5} \cdot i = 4i \sqrt{5}
\][/tex]
4. Add to the real part: Lastly, combine this with the given real number 6:
[tex]\[
6 + \sqrt{-80} = 6 + 4i \sqrt{5}
\][/tex]
Thus, the simplified form of the expression [tex]\(6 + \sqrt{-80}\)[/tex] is:
[tex]\[
6 + 4i \sqrt{5}
\][/tex]
