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Sagot :
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]
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]
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