To simplify the expression [tex]\(\frac{4+\sqrt{-121}}{2(3)}\)[/tex], let's follow through the steps carefully:
1. Understand the Square Root of a Negative Number:
- The term [tex]\(\sqrt{-121}\)[/tex] involves a square root of a negative number, which deals with imaginary numbers. By definition, [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
- We know that [tex]\(\sqrt{-121}\)[/tex] can be rewritten as [tex]\(\sqrt{121 \cdot -1}\)[/tex].
- This simplifies to [tex]\(\sqrt{121} \cdot \sqrt{-1}\)[/tex].
- Given [tex]\(\sqrt{121} = 11\)[/tex] and [tex]\(\sqrt{-1} = i\)[/tex], we have [tex]\(\sqrt{-121} = 11i\)[/tex].
2. Substitute the Imaginary Component:
- Substitute [tex]\(\sqrt{-121}\)[/tex] with [tex]\(11i\)[/tex] in the expression: [tex]\(\frac{4 + 11i}{2(3)}\)[/tex].
3. Simplify the Denominator:
- The denominator [tex]\(2(3)\)[/tex] equals [tex]\(6\)[/tex].
4. Break Down the Fraction:
- Now, the expression can be written as [tex]\(\frac{4 + 11i}{6}\)[/tex].
- This can be separated into two distinct fractions: [tex]\(\frac{4}{6} + \frac{11i}{6}\)[/tex].
5. Simplify Each Term:
- Simplify [tex]\(\frac{4}{6}\)[/tex]:
- [tex]\(\frac{4}{6} = \frac{2}{3}\)[/tex].
- Simplify [tex]\(\frac{11i}{6}\)[/tex]:
- [tex]\(\frac{11i}{6}\)[/tex] is already in its simplest form.
6. Combine the Simplified Terms:
- Therefore, combining the simplified forms we get: [tex]\(\frac{2}{3} + \frac{11i}{6}\)[/tex].
Thus, the simplified form of the expression [tex]\(\frac{4 + \sqrt{-121}}{2(3)}\)[/tex] is:
[tex]\[
\frac{2}{3} + \frac{11i}{6} \approx (0.6667 + 1.8333i)
\][/tex]
Where [tex]\(0.6667\)[/tex] is the decimal representation of [tex]\(\frac{2}{3}\)[/tex] and [tex]\(1.8333\)[/tex] is the decimal representation of [tex]\(\frac{11}{6}\)[/tex], for better understanding the numerical relationship.
