Sure! Let's work through the given expression step-by-step to simplify it.
We start with the given expression:
[tex]\[ x = \frac{5 + \sqrt{-49}}{6} \][/tex]
1. Identify the square root of the negative number:
The square root of [tex]\(-49\)[/tex] can be written in terms of imaginary numbers. Specifically:
[tex]\[ \sqrt{-49} = \sqrt{-1 \cdot 49} = \sqrt{-1} \cdot \sqrt{49} = i \cdot 7 = 7i \][/tex]
Here, [tex]\( i \)[/tex] represents the imaginary unit where [tex]\( i^2 = -1 \)[/tex].
2. Substitute the value of [tex]\(\sqrt{-49}\)[/tex] into the expression:
[tex]\[ x = \frac{5 + 7i}{6} \][/tex]
3. Separate the expression into real and imaginary parts:
We can split the numerator and distribute the division over the addition:
[tex]\[ x = \frac{5}{6} + \frac{7i}{6} \][/tex]
Therefore, the expression can be written as:
[tex]\[ x = \frac{5}{6} + \frac{7i}{6} \][/tex]
This is the standard form of a complex number, where [tex]\(\frac{5}{6}\)[/tex] is the real part and [tex]\(\frac{7}{6}\)[/tex] is the imaginary part.
4. List possible solutions based on the form [tex]\(\frac{a}{b} + \frac{c}{d} i\)[/tex]:
The simplified form of the given expression is:
[tex]\[ x = \frac{5}{6} + \frac{7i}{6} \][/tex]
Given this expression, we can match it against the possible choices provided:
[tex]\[
\begin{array}{c}
\text{-} \frac{1}{3} i \\
2 i \\
\frac{5}{6} + \frac{7i}{6} \\
\frac{5}{6} - \frac{7i}{6}
\end{array}
\][/tex]
The correct match is:
[tex]\[ \frac{5}{6} + \frac{7i}{6} \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \boxed{\frac{5}{6} + \frac{7i}{6}} \][/tex]
