Answer :
To simplify the given expression [tex]\(\frac{1+\sqrt{-18}}{-1-\sqrt{-2}}\)[/tex], follow these steps:
1. Simplify inside the square roots:
- Recognize that [tex]\(\sqrt{-18}\)[/tex] can be written as [tex]\(\sqrt{18} \cdot i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Thus, [tex]\(\sqrt{-18} = \sqrt{18} \cdot i = 3\sqrt{2} \cdot i\)[/tex].
- Similarly, [tex]\(\sqrt{-2} = \sqrt{2} \cdot i\)[/tex].
2. Rewrite the original expression with these substitutions:
[tex]\[ \frac{1 + 3\sqrt{2} \cdot i}{-1 - \sqrt{2} \cdot i} \][/tex]
3. Rationalize the denominator:
- To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(-1 - \sqrt{2} \cdot i\)[/tex] is [tex]\(-1 + \sqrt{2} \cdot i\)[/tex].
- This gives:
[tex]\[ \frac{(1 + 3\sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i)}{(-1 - \sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i)} \][/tex]
4. Multiply the numerator:
- Expand the numerator using the distributive property (FOIL method):
[tex]\[ (1 + 3\sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i) = 1 \cdot (-1) + 1 \cdot (\sqrt{2} \cdot i) + (3\sqrt{2} \cdot i) \cdot (-1) + (3\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) \][/tex]
- Simplify each term:
[tex]\[ 1 \cdot (-1) = -1 \][/tex]
[tex]\[ 1 \cdot (\sqrt{2} \cdot i) = \sqrt{2} \cdot i \][/tex]
[tex]\[ (3\sqrt{2} \cdot i) \cdot (-1) = -3\sqrt{2} \cdot i \][/tex]
[tex]\[ (3\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) = 3\sqrt{2} \cdot \sqrt{2} \cdot i^2 = 3 \cdot 2 \cdot (-1) = -6 \][/tex]
- Combine all terms:
[tex]\[ -1 + \sqrt{2} \cdot i - 3\sqrt{2} \cdot i - 6 = -1 - 6 + (\sqrt{2} - 3\sqrt{2}) \cdot i = -7 - 2\sqrt{2} \cdot i \][/tex]
5. Multiply the denominator:
- Expand and simplify the denominator:
[tex]\[ (-1 - \sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i) = (-1) \cdot (-1) + (-1) \cdot (\sqrt{2} \cdot i) + (-\sqrt{2} \cdot i) \cdot (-1) + (-\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) \][/tex]
- Simplify each term:
[tex]\[ (-1) \cdot (-1) = 1 \][/tex]
[tex]\[ (-1) \cdot (\sqrt{2} \cdot i) = -\sqrt{2} \cdot i \][/tex]
[tex]\[ (-\sqrt{2} \cdot i) \cdot (-1) = \sqrt{2} \cdot i \][/tex]
[tex]\[ (-\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) = -2 i^2 = 2 \][/tex]
- Combine all terms:
[tex]\[ 1 - \sqrt{2} \cdot i + \sqrt{2} \cdot i + 2 = 1 + 2 = 3 \][/tex]
6. Combine the results:
- The numerator is [tex]\(-7 - 2\sqrt{2} \cdot i\)[/tex].
- The denominator is [tex]\(3\)[/tex].
- Thus, the expression becomes:
[tex]\[ \frac{-7 - 2\sqrt{2} \cdot i}{3} = \frac{-7}{3} + \frac{-2\sqrt{2} \cdot i}{3} \][/tex]
7. Simplify further (if needed):
- Let's re-arrange it into a form which looks more simplified:
- Since we know the result is a more simplified, rationalized form, we can condense the numerator and denominator.
[tex]\[ \frac{1 + 3\sqrt{2} \cdot i}{-1 - \sqrt{2} \cdot i} = \frac{(-3\sqrt{2} + i)}{\sqrt{2} - i} \][/tex]
Hence the final simplified form of the given expression is:
[tex]\[ \boxed{\frac{-3\sqrt{2} + i}{\sqrt{2} - i}} \][/tex]
1. Simplify inside the square roots:
- Recognize that [tex]\(\sqrt{-18}\)[/tex] can be written as [tex]\(\sqrt{18} \cdot i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Thus, [tex]\(\sqrt{-18} = \sqrt{18} \cdot i = 3\sqrt{2} \cdot i\)[/tex].
- Similarly, [tex]\(\sqrt{-2} = \sqrt{2} \cdot i\)[/tex].
2. Rewrite the original expression with these substitutions:
[tex]\[ \frac{1 + 3\sqrt{2} \cdot i}{-1 - \sqrt{2} \cdot i} \][/tex]
3. Rationalize the denominator:
- To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(-1 - \sqrt{2} \cdot i\)[/tex] is [tex]\(-1 + \sqrt{2} \cdot i\)[/tex].
- This gives:
[tex]\[ \frac{(1 + 3\sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i)}{(-1 - \sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i)} \][/tex]
4. Multiply the numerator:
- Expand the numerator using the distributive property (FOIL method):
[tex]\[ (1 + 3\sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i) = 1 \cdot (-1) + 1 \cdot (\sqrt{2} \cdot i) + (3\sqrt{2} \cdot i) \cdot (-1) + (3\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) \][/tex]
- Simplify each term:
[tex]\[ 1 \cdot (-1) = -1 \][/tex]
[tex]\[ 1 \cdot (\sqrt{2} \cdot i) = \sqrt{2} \cdot i \][/tex]
[tex]\[ (3\sqrt{2} \cdot i) \cdot (-1) = -3\sqrt{2} \cdot i \][/tex]
[tex]\[ (3\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) = 3\sqrt{2} \cdot \sqrt{2} \cdot i^2 = 3 \cdot 2 \cdot (-1) = -6 \][/tex]
- Combine all terms:
[tex]\[ -1 + \sqrt{2} \cdot i - 3\sqrt{2} \cdot i - 6 = -1 - 6 + (\sqrt{2} - 3\sqrt{2}) \cdot i = -7 - 2\sqrt{2} \cdot i \][/tex]
5. Multiply the denominator:
- Expand and simplify the denominator:
[tex]\[ (-1 - \sqrt{2} \cdot i)(-1 + \sqrt{2} \cdot i) = (-1) \cdot (-1) + (-1) \cdot (\sqrt{2} \cdot i) + (-\sqrt{2} \cdot i) \cdot (-1) + (-\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) \][/tex]
- Simplify each term:
[tex]\[ (-1) \cdot (-1) = 1 \][/tex]
[tex]\[ (-1) \cdot (\sqrt{2} \cdot i) = -\sqrt{2} \cdot i \][/tex]
[tex]\[ (-\sqrt{2} \cdot i) \cdot (-1) = \sqrt{2} \cdot i \][/tex]
[tex]\[ (-\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) = -2 i^2 = 2 \][/tex]
- Combine all terms:
[tex]\[ 1 - \sqrt{2} \cdot i + \sqrt{2} \cdot i + 2 = 1 + 2 = 3 \][/tex]
6. Combine the results:
- The numerator is [tex]\(-7 - 2\sqrt{2} \cdot i\)[/tex].
- The denominator is [tex]\(3\)[/tex].
- Thus, the expression becomes:
[tex]\[ \frac{-7 - 2\sqrt{2} \cdot i}{3} = \frac{-7}{3} + \frac{-2\sqrt{2} \cdot i}{3} \][/tex]
7. Simplify further (if needed):
- Let's re-arrange it into a form which looks more simplified:
- Since we know the result is a more simplified, rationalized form, we can condense the numerator and denominator.
[tex]\[ \frac{1 + 3\sqrt{2} \cdot i}{-1 - \sqrt{2} \cdot i} = \frac{(-3\sqrt{2} + i)}{\sqrt{2} - i} \][/tex]
Hence the final simplified form of the given expression is:
[tex]\[ \boxed{\frac{-3\sqrt{2} + i}{\sqrt{2} - i}} \][/tex]
