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Original Question
√-10 in simplest radical form.Question
√-75 in simplest radical form.
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Answered by
GPT-4o mini
To simplify \(\sqrt{-75}\), we can separate the negative sign from the square root. We know that \(\sqrt{-1} = i\), where \(i\) is the imaginary unit.
Thus, we can rewrite \(\sqrt{-75}\) as:
\[
\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1} = \sqrt{75} \cdot i
\]
Next, we simplify \(\sqrt{75}\). The number 75 can be factored into prime factors:
\[
75 = 25 \times 3 = 5^2 \times 3
\]
Taking the square root, we have:
\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
\]
Now, substituting this back into our expression for \(\sqrt{-75}\):
\[
\sqrt{-75} = 5\sqrt{3} \cdot i
\]
Therefore, the simplest radical form of \(\sqrt{-75}\) is:
\[
5\sqrt{3} i
\]
Thus, we can rewrite \(\sqrt{-75}\) as:
\[
\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1} = \sqrt{75} \cdot i
\]
Next, we simplify \(\sqrt{75}\). The number 75 can be factored into prime factors:
\[
75 = 25 \times 3 = 5^2 \times 3
\]
Taking the square root, we have:
\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
\]
Now, substituting this back into our expression for \(\sqrt{-75}\):
\[
\sqrt{-75} = 5\sqrt{3} \cdot i
\]
Therefore, the simplest radical form of \(\sqrt{-75}\) is:
\[
5\sqrt{3} i
\]
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