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Simplify the expression:

[tex]\[ \frac{48 - \sqrt{-448}}{8} \][/tex]

Sagot :

GinnyAnswer
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]
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