To simplify the given expression [tex]\(\frac{\sqrt{-10} \cdot \sqrt{-20}}{1-\sqrt{-4}}\)[/tex], let's proceed step-by-step:
### Step 1: Simplify the Numerator
The numerator of the expression is [tex]\(\sqrt{-10} \cdot \sqrt{-20}\)[/tex].
1. Simplify [tex]\(\sqrt{-10}\)[/tex]:
[tex]\[
\sqrt{-10} = \sqrt{10} \cdot i
\][/tex]
where [tex]\(i\)[/tex] is the imaginary unit.
2. Simplify [tex]\(\sqrt{-20}\)[/tex]:
[tex]\[
\sqrt{-20} = \sqrt{20} \cdot i
\][/tex]
3. Multiply [tex]\(\sqrt{10} \cdot i\)[/tex] and [tex]\(\sqrt{20} \cdot i\)[/tex]:
[tex]\[
\sqrt{-10} \cdot \sqrt{-20} = (\sqrt{10} \cdot i) \cdot (\sqrt{20} \cdot i) = \sqrt{10} \cdot \sqrt{20} \cdot i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], the expression simplifies to:
[tex]\[
\sqrt{10} \cdot \sqrt{20} \cdot (-1)
\][/tex]
[tex]\[
= -\sqrt{10 \cdot 20}
\][/tex]
[tex]\[
= -\sqrt{200}
\][/tex]
Since [tex]\(\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}\)[/tex], we have:
[tex]\[
-\sqrt{200} = -10\sqrt{2}
\][/tex]
Numerically, [tex]\(-10\sqrt{2} \approx -14.142135623730953\)[/tex].
### Step 2: Simplify the Denominator
The denominator of the expression is [tex]\(1 - \sqrt{-4}\)[/tex].
1. Simplify [tex]\(\sqrt{-4}\)[/tex]:
[tex]\[
\sqrt{-4} = \sqrt{4} \cdot i = 2i
\][/tex]
2. Form the whole denominator:
[tex]\[
1 - \sqrt{-4} = 1 - 2i
\][/tex]
### Step 3: Divide the Numerator by the Denominator
Now we need to divide [tex]\(-10\sqrt{2}\)[/tex] by [tex]\(1 - 2i\)[/tex].
Performing this division in complex number arithmetic, in rectangular form, we get:
The numerator is [tex]\(-14.142135623730953\)[/tex].
The denominator is [tex]\(1 - 2i\)[/tex].
Simplifying the division:
[tex]\[
\frac{-14.142135623730953}{1-2i} \approx -2.8284271247461907 - 5.6568542494923815i
\][/tex]
### Conclusion
The simplified result of the given expression [tex]\(\frac{\sqrt{-10} \cdot \sqrt{-20}}{1 - \sqrt{-4}}\)[/tex] is:
[tex]\[
-2.8284271247461907 - 5.6568542494923815i
\][/tex]
