To determine the value of [tex]\(\sqrt{-3} \sqrt{-27}\)[/tex], let's break down the problem step-by-step.
1. Calculate [tex]\(\sqrt{-3}\)[/tex]:
- Remember that the square root of a negative number involves an imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
- [tex]\(\sqrt{-3}\)[/tex] can be expressed as [tex]\(\sqrt{3} \cdot \sqrt{-1} = \sqrt{3} \cdot i\)[/tex].
- In our case, [tex]\(\sqrt{3} \cdot i = 1.0605752387249068e-16 + 1.7320508075688772j\)[/tex].
2. Calculate [tex]\(\sqrt{-27}\)[/tex]:
- Similarly, [tex]\(\sqrt{-27}\)[/tex] can be expressed as [tex]\(\sqrt{27} \cdot \sqrt{-1} = \sqrt{27} \cdot i\)[/tex].
- [tex]\(\sqrt{27} \cdot i = 3.181725716174721e-16 + 5.196152422706632j\)[/tex].
3. Multiply [tex]\(\sqrt{-3}\)[/tex] and [tex]\(\sqrt{-27}\)[/tex]:
- When we multiply two imaginary numbers, we follow the rule: [tex]\(\sqrt{a} \cdot i \cdot \sqrt{b} \cdot i = \sqrt{a \cdot b} \cdot i^2\)[/tex].
- For [tex]\(a = 3\)[/tex] and [tex]\(b = 27\)[/tex], we get [tex]\(\sqrt{3 \cdot 27} \cdot i^2 = \sqrt{81} \cdot (-1) = -9\)[/tex].
Therefore, the value of [tex]\(\sqrt{-3} \sqrt{-27}\)[/tex] is [tex]\(-9\)[/tex].
Among the given options:
(a) 9
(b) -9
(c) [tex]\(\sqrt{-81}\)[/tex]
(d) [tex]\(\pm 9\)[/tex]
The correct answer is:
(b) -9
