Let's simplify the expression [tex]\(6 + \sqrt{-80}\)[/tex] step-by-step.
1. Identify the square root of a negative number:
[tex]\[
\sqrt{-80}
\][/tex]
To deal with the square root of a negative number, we use the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Thus, we can rewrite [tex]\(\sqrt{-80}\)[/tex] as:
[tex]\[
\sqrt{-80} = \sqrt{80} \cdot \sqrt{-1} = \sqrt{80} \cdot i
\][/tex]
2. Simplify the square root of 80:
Let's break down 80 into its prime factors:
[tex]\[
80 = 16 \cdot 5
\][/tex]
Now, we can use the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], which gives us:
[tex]\[
\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5}
\][/tex]
3. Combine the results from the previous steps:
[tex]\[
\sqrt{-80} = 4 \sqrt{5} \cdot i
\][/tex]
4. Add this to the original expression:
[tex]\[
6 + \sqrt{-80} = 6 + 4 \sqrt{5} \cdot i
\][/tex]
Thus, the simplified form of [tex]\(6 + \sqrt{-80}\)[/tex] is:
[tex]\[
6 + 4 \sqrt{5}i
\][/tex]
