Which of the following is equivalent to `(10-sqrt(-12))/(1-sqrt(-27))`? (Note: `i=sqrt(-1)`)

Options

A. `(-2)/(7)`
B. `(28)/(3)`
C. `(-2)/(7)+isqrt(3)`
D. `1+isqrt(3)`

Answer

To solve the expression \((10 - \sqrt{-12}) / (1 - \sqrt{-27})\), we will follow these steps:

### Step 1: Rewrite the square roots of negative numbers using \(i\)
We know that \(i = \sqrt{-1}\). Therefore, we can rewrite the square roots in the expression:

\[ \sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i \]
\[ \sqrt{-27} = \sqrt{27} \cdot i = 3\sqrt{3} \cdot i \]

Substituting these into the expression gives us:

\[ \frac{10 - 2\sqrt{3}i}{1 - 3\sqrt{3}i} \]

### Step 2: Multiply the numerator and denominator by the conjugate of the denominator
To eliminate the imaginary unit \(i\) from the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(1 + 3\sqrt{3}i\):

\[ \frac{(10 - 2\sqrt{3}i)(1 + 3\sqrt{3}i)}{(1 - 3\sqrt{3}i)(1 + 3\sqrt{3}i)} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \(a^2 - b^2\):

\[ (1)^2 - (3\sqrt{3}i)^2 = 1 - 27(-1) = 1 + 27 = 28 \]

### Step 4: Simplify the numerator
Now we will expand the numerator:

\[ (10 - 2\sqrt{3}i)(1 + 3\sqrt{3}i) = 10 \cdot 1 + 10 \cdot 3\sqrt{3}i - 2\sqrt{3}i \cdot 1 - 2\sqrt{3}i \cdot 3\sqrt{3}i \]

Calculating each term:

- \(10 \cdot 1 = 10\)
- \(10 \cdot 3\sqrt{3}i = 30\sqrt{3}i\)
- \(-2\sqrt{3}i \cdot 1 = -2\sqrt{3}i\)
- \(-2\sqrt{3}i \cdot 3\sqrt{3}i = -6 \cdot 3(-1) = 18\)

Combining these gives:

\[ 10 + 30\sqrt{3}i - 2\sqrt{3}i + 18 = 28 + 28\sqrt{3}i \]

### Step 5: Combine the results
Now we can write the full expression:

\[ \frac{28 + 28\sqrt{3}i}{28} \]

### Step 6: Simplify the final expression
Dividing each term by 28:

\[ 1 + \sqrt{3}i \]

Thus, the expression \((10 - \sqrt{-12}) / (1 - \sqrt{-27})\) simplifies to:

\[ \boxed{1 + \sqrt{3}i} \]

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