Question

express the radical using the imaginary unit, i. express your answer in simplified form. ±√(-25) = ±

Explanation:

Step1: Recall the definition of \(i\)

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). So we can rewrite \(\sqrt{-25}\) as \(\sqrt{25\times(-1)}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0\) and \(b\geq0\), here we extend it to complex numbers), we have \(\sqrt{25\times(-1)}=\sqrt{25}\times\sqrt{-1}\).

Step3: Simplify the square roots

We know that \(\sqrt{25} = 5\) and \(\sqrt{-1}=i\), so \(\sqrt{25}\times\sqrt{-1}=5\times i = 5i\). And since we have \(\pm\sqrt{-25}\), the result is \(\pm5i\).

Answer:

\(5i\) (and also \(-5i\), but in the box for the simplified form of \(\pm\sqrt{-25}\) we write \(\pm5i\), so the answer inside the box is \(5i\) considering the \(\pm\) outside, but following the problem's request to express the radical (the part inside the \(\pm\)) in simplified form with \(i\), the answer is \(5i\) (with the \(\pm\) already given). So the final answer is \(\pm5i\), and the simplified form of the radical (the \(\sqrt{-25}\) part) is \(5i\) (with the \(\pm\) outside), so the answer to put in the box is \(5i\) (and the overall is \(\pm5i\)).