Question

use the imaginary number i to rewrite the expression below as a com simplify all radicals. \\(sqrt{-25}\\)

Explanation:

Step1: Recall the definition of imaginary unit

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 \( a\geq0, b\geq0 \), and 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 substituting these values in, we get \( 5\times i = 5i \).

Answer:

\( 5i \)