To solve the expression [tex]\(\frac{\sqrt{-2}}{\sqrt{-8}}\)[/tex], let's break it down step by step:
1. Understanding Square Roots of Negative Numbers:
- The square root of a negative number leads us to the concept of complex numbers. The square root of [tex]\(-1\)[/tex] is denoted as [tex]\(i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
- Therefore, [tex]\(\sqrt{-a} = \sqrt{a} \cdot i\)[/tex] for any positive real number [tex]\(a\)[/tex].
2. Calculate [tex]\(\sqrt{-2}\)[/tex]:
- Rewrite [tex]\(\sqrt{-2}\)[/tex] in terms of the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-2} = \sqrt{2} \cdot i
\][/tex]
3. Calculate [tex]\(\sqrt{-8}\)[/tex]:
- Rewrite [tex]\(\sqrt{-8}\)[/tex] in terms of the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-8} = \sqrt{8} \cdot i
\][/tex]
- Simplify [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\][/tex]
- Therefore,
[tex]\[
\sqrt{-8} = 2\sqrt{2} \cdot i
\][/tex]
4. Substitute These Values into the Expression:
- Now, substitute the values of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-8}\)[/tex] into the expression:
[tex]\[
\frac{\sqrt{-2}}{\sqrt{-8}} = \frac{\sqrt{2} \cdot i}{2\sqrt{2} \cdot i}
\][/tex]
5. Simplify the Expression:
- Cancel out the common terms:
[tex]\[
\frac{\sqrt{2} \cdot i}{2\sqrt{2} \cdot i} = \frac{1}{2} = 0.5
\][/tex]
- Since [tex]\(i\)[/tex] cancels out, the imaginary part becomes zero, leaving us with a real number.
6. Combine the Results:
- The final result of the division of the square roots is:
[tex]\[
\frac{\sqrt{-2}}{\sqrt{-8}} = 0.5 + 0j
\][/tex]
7. Summary:
- [tex]\(\sqrt{-2}\)[/tex] evaluates to [tex]\((8.659560562354934e-17 + 1.4142135623730951j)\)[/tex]
- [tex]\(\sqrt{-8}\)[/tex] evaluates to [tex]\((1.7319121124709868e-16 + 2.8284271247461903j)\)[/tex]
- The division [tex]\(\frac{\sqrt{-2}}{\sqrt{-8}} = (0.5 + 0j)\)[/tex]
So, [tex]\(\frac{\sqrt{-2}}{\sqrt{-8}} = 0.5\)[/tex].
