Certainly! Let's solve the given expression step by step:
[tex]\[
(5 - 3\sqrt{-48})(2 - 4\sqrt{-27})
\][/tex]
### Step 1: Simplify the square root of the negative numbers
First, we recognize that the square roots of negative numbers result in imaginary numbers. Specifically, we use [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
[tex]\[
\sqrt{-48} = \sqrt{48} \cdot i \quad \text{and} \quad \sqrt{-27} = \sqrt{27} \cdot i
\][/tex]
### Step 2: Calculate the square roots of the positive parts
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}
\][/tex]
[tex]\[
\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}
\][/tex]
### Step 3: Substitute back into the expression
Using the imaginary unit, [tex]\(i\)[/tex], we get:
[tex]\[
\sqrt{-48} = 4\sqrt{3}i \quad \text{and} \quad \sqrt{-27} = 3\sqrt{3}i
\][/tex]
So the original expression becomes:
[tex]\[
(5 - 3 \cdot 4\sqrt{3}i)(2 - 4 \cdot 3\sqrt{3}i)
\][/tex]
[tex]\[
(5 - 12\sqrt{3}i)(2 - 12\sqrt{3}i)
\][/tex]
### Step 4: Expand the product using the distributive property
[tex]\[
(5 - 12\sqrt{3}i)(2 - 12\sqrt{3}i) = 5 \cdot 2 + 5 \cdot (-12\sqrt{3}i) + (-12\sqrt{3}i) \cdot 2 + (-12\sqrt{3}i) \cdot (-12\sqrt{3}i)
\][/tex]
[tex]\[
= 10 - 60\sqrt{3}i - 24\sqrt{3}i + 144(\sqrt{3}i)^2
\][/tex]
### Step 5: Simplify the expression
Combine like terms and remember that [tex]\((i)^2 = -1\)[/tex]:
[tex]\[
= 10 - 84\sqrt{3}i + 144 \cdot 3 \cdot (-1)
\][/tex]
[tex]\[
= 10 - 84\sqrt{3}i - 432
\][/tex]
[tex]\[
= -422 - 84\sqrt{3}i
\][/tex]
### Numerical computation
Note: [tex]\(\sqrt{3}\)[/tex] approximately equals 1.732. Thus,
[tex]\[
84\sqrt{3} \approx 84 \cdot 1.732 = 145.492
\][/tex]
So the final result is:
[tex]\[
-422 - 145.492i
\][/tex]
Thus, the answer to the expression [tex]\((5 - 3 \sqrt{-48})(2 - 4 \sqrt{-27})\)[/tex] is given by:
[tex]\[
(-422 - 145.492i)
\][/tex]
