Sure, let's go through the solution step-by-step for the given expression:
[tex]\[
\frac{48 - \sqrt{-448}}{8}
\][/tex]
### Step 1: Simplify the Square Root
The expression inside the square root is [tex]\(-448\)[/tex]. We need to extract the square root of a negative number, which involves imaginary numbers. Recall that:
[tex]\[
\sqrt{-1} = i \quad \text{where } i \text{ is the imaginary unit.}
\][/tex]
Therefore:
[tex]\[
\sqrt{-448} = \sqrt{448} \cdot \sqrt{-1} = \sqrt{448} \cdot i
\][/tex]
### Step 2: Simplify [tex]\(\sqrt{448}\)[/tex]
We can further simplify [tex]\(\sqrt{448}\)[/tex]:
[tex]\[
448 = 64 \times 7 \quad \Rightarrow \quad \sqrt{448} = \sqrt{64 \times 7} = \sqrt{64} \cdot \sqrt{7} = 8\sqrt{7}
\][/tex]
Hence:
[tex]\[
\sqrt{-448} = 8\sqrt{7} \cdot i
\][/tex]
### Step 3: Substitute Back into the Original Expression
Now substitute [tex]\(\sqrt{-448} = 8\sqrt{7} \cdot i\)[/tex] back into the original expression:
[tex]\[
\frac{48 - 8\sqrt{7} \cdot i}{8}
\][/tex]
### Step 4: Simplify the Fraction
Divide both parts in the numerator by 8:
[tex]\[
\frac{48}{8} - \frac{8\sqrt{7} \cdot i}{8} = 6 - \sqrt{7} \cdot i
\][/tex]
### Step 5: Numerical Values
In this case, the numerical value for [tex]\(\sqrt{7}\)[/tex] is approximately 2.6457513110645907. Therefore,
[tex]\[
6 - \sqrt{7} \cdot i \approx 6 - 2.6457513110645907i
\][/tex]
### Final Answer
The simplified and evaluated form of the given expression is:
[tex]\[
\frac{48 - \sqrt{-448}}{8} = 6 - 2.6457513110645907i
\][/tex]
