To solve the expression [tex]\(\sqrt{-108} - \sqrt{-3}\)[/tex], we need to work with complex numbers, as the square roots of negative numbers are not real.
1. Calculate [tex]\(\sqrt{-108}\)[/tex]:
Express [tex]\(\sqrt{-108}\)[/tex] in terms of [tex]\(i\)[/tex] (the imaginary unit, where [tex]\(i = \sqrt{-1}\)[/tex]):
[tex]\[
\sqrt{-108} = \sqrt{108 \cdot (-1)} = \sqrt{108} \cdot i
\][/tex]
We know that:
[tex]\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}
\][/tex]
Thus:
[tex]\[
\sqrt{-108} = 6\sqrt{3} \cdot i
\][/tex]
Converting [tex]\(6\sqrt{3}\)[/tex] into a numerical decimal value:
[tex]\[
6\sqrt{3} \approx 10.392304845413264
\][/tex]
Therefore:
[tex]\[
\sqrt{-108} \approx 10.392304845413264i
\][/tex]
2. Calculate [tex]\(\sqrt{-3}\)[/tex]:
Similarly, express [tex]\(\sqrt{-3}\)[/tex] in terms of [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-3} = \sqrt{3 \cdot (-1)} = \sqrt{3} \cdot i
\][/tex]
Using the decimal value for [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\sqrt{3} \approx 1.7320508075688772
\][/tex]
Therefore:
[tex]\[
\sqrt{-3} \approx 1.7320508075688772i
\][/tex]
3. Subtract [tex]\(\sqrt{-3}\)[/tex] from [tex]\(\sqrt{-108}\)[/tex]:
We now have:
[tex]\[
\sqrt{-108} - \sqrt{-3} = 10.392304845413264i - 1.7320508075688772i
\][/tex]
Combine the imaginary components:
[tex]\[
10.392304845413264i - 1.7320508075688772i = (10.392304845413264 - 1.7320508075688772)i
\][/tex]
Evaluate the subtraction:
[tex]\[
10.392304845413264 - 1.7320508075688772 = 8.660254037844387
\][/tex]
So, the result is:
[tex]\[
8.660254037844387i
\][/tex]
Putting it all together, the expression [tex]\(\sqrt{-108} - \sqrt{-3}\)[/tex] is equivalent to:
[tex]\[
(5.302876193624535e-16 + 8.660254037844387i)
\][/tex]
