Express $(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-\sqrt{-1})$ into a + ib form

Q: Express $(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-\sqrt{-1})$ into a + ib form

To express the expression $(- \sqrt{3} + \sqrt{-2})(2\sqrt{3} - \sqrt{-1})$ in the form $a + ib$, we need to simplify it step by step. First, let's rewrite the expression using imaginary numbers: - $\sqrt{-2}$ can be written as $i\sqrt{2}$. - $\sqrt{-1}$ is $i$. Thus, the expression becomes: \[ (-\sqrt{3} + i\sqrt{2})(2\sqrt{3} - i) \] Now, we will expand this expression using the distributive property (FOIL method): 1. Multiply the first terms: \[ (-\sqrt{3}) \cdot (2\sqrt{3}) = -2 \cdot 3 = -6 \] 2. Multiply the outer terms: \[ (-\sqrt{3}) \cdot (-i) = \sqrt{3}i \] 3. Multiply the inner terms: \[ (i\sqrt{2}) \cdot (2\sqrt{3}) = 2i\sqrt{6} \] 4. Multiply the last terms: \[ (i\sqrt{2}) \cdot (-i) = -i^2\sqrt{2} = \sqrt{2} \quad (\text{since } i^2 = -1) \] Now, combine all these results: \[ -6 + \sqrt{3}i + 2i\sqrt{6} + \sqrt{2} \] Combine the real parts and the imaginary parts: - Real part: $-6 + \sqrt{2}$ - Imaginary part: $\sqrt{3}i + 2i\sqrt{6} = i(\sqrt{3} + 2\sqrt{6})$ Thus, the expression in $a + ib$ form is: \[ (-6 + \sqrt{2}) + i(\sqrt{3} + 2\sqrt{6}) \] This matches Option 1: $(-6+\sqrt{2})+\sqrt{3}(1+2\sqrt{2})i$. Now, let's explain why the other options are incorrect: - **Option 2**: $(+6-\sqrt{2})+\sqrt{3}(1+2\sqrt{2})i$ - The real part here is $+6-\sqrt{2}$, which does not match the calculated real part $-6+\sqrt{2}$. - **Option 3**: $( -6+\sqrt{2})-\sqrt{3}(1+2\sqrt{2})i$ - The imaginary part here is negative, $-\sqrt{3}(1+2\sqrt{2})i$, which contradicts the positive imaginary part $\sqrt{3}(1+2\sqrt{2})i$. - **Option 4**: $( -6+\sqrt{2})+\sqrt{3}(1-2\sqrt{2})i$ - The imaginary part here is $\sqrt{3}(1-2\sqrt{2})i$, which does not match the calculated imaginary part $\sqrt{3}(1+2\sqrt{2})i$. Therefore, Option 1 is the correct answer, as it accurately represents the expression in the form $a + ib$.

Question

Accepted Answer

To express the expression $(- \sqrt{3} + \sqrt{-2})(2\sqrt{3} - \sqrt{-1})$ in the form $a + ib$, we need to simplify it step by step.

First, let's rewrite the expression using imaginary numbers:
- $\sqrt{-2}$ can be written as $i\sqrt{2}$.
- $\sqrt{-1}$ is $i$.

Thus, the expression becomes:
$$
(-\sqrt{3} + i\sqrt{2})(2\sqrt{3} - i)
$$

Now, we will expand this expression using the distributive property (FOIL method):

1. Multiply the first terms: 
   $$
   (-\sqrt{3}) \cdot (2\sqrt{3}) = -2 \cdot 3 = -6
   $$

2. Multiply the outer terms:
   $$
   (-\sqrt{3}) \cdot (-i) = \sqrt{3}i
   $$

3. Multiply the inner terms:
   $$
   (i\sqrt{2}) \cdot (2\sqrt{3}) = 2i\sqrt{6}
   $$

4. Multiply the last terms:
   $$
   (i\sqrt{2}) \cdot (-i) = -i^2\sqrt{2} = \sqrt{2} \quad (\text{since } i^2 = -1)
   $$

Now, combine all these results:
$$
-6 + \sqrt{3}i + 2i\sqrt{6} + \sqrt{2}
$$

Combine the real parts and the imaginary parts:
- Real part: $-6 + \sqrt{2}$
- Imaginary part: $\sqrt{3}i + 2i\sqrt{6} = i(\sqrt{3} + 2\sqrt{6})$

Thus, the expression in $a + ib$ form is:
$$
(-6 + \sqrt{2}) + i(\sqrt{3} + 2\sqrt{6})
$$

This matches Option 1: $(-6+\sqrt{2})+\sqrt{3}(1+2\sqrt{2})i$.

Now, let's explain why the other options are incorrect:

- **Option 2**: $(+6-\sqrt{2})+\sqrt{3}(1+2\sqrt{2})i$
  - The real part here is $+6-\sqrt{2}$, which does not match the calculated real part $-6+\sqrt{2}$.

- **Option 3**: $( -6+\sqrt{2})-\sqrt{3}(1+2\sqrt{2})i$
  - The imaginary part here is negative, $-\sqrt{3}(1+2\sqrt{2})i$, which contradicts the positive imaginary part $\sqrt{3}(1+2\sqrt{2})i$.

- **Option 4**: $( -6+\sqrt{2})+\sqrt{3}(1-2\sqrt{2})i$
  - The imaginary part here is $\sqrt{3}(1-2\sqrt{2})i$, which does not match the calculated imaginary part $\sqrt{3}(1+2\sqrt{2})i$.

Therefore, Option 1 is the correct answer, as it accurately represents the expression in the form $a + ib$.