Question

simplify roots of negative numbers
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express the radical using the
imaginary unit, i.
express your answer in simplified form.
±√(-37) = ±
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simplifying roots of
negative numbers

Explanation:

Step1: Recall the definition of imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-a} \) (where \( a>0 \)) as \( \sqrt{a}\cdot\sqrt{-1} \).
For \( \pm\sqrt{-37} \), we can split the square root: \( \pm\sqrt{-37}=\pm\sqrt{37\times(-1)} \)

Step2: Apply the property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0, b\geq0 \); here we extend it to complex numbers where \( b = - 1\)), we get \( \pm\sqrt{37\times(-1)}=\pm\sqrt{37}\cdot\sqrt{-1} \)
Since \( \sqrt{-1}=i \), we substitute it in: \( \pm\sqrt{37}\cdot i=\pm i\sqrt{37} \)

Answer:

\( i\sqrt{37} \) (and also \( -i\sqrt{37} \), but combined as \( \pm i\sqrt{37} \))