Problem 46 - Introductory Algebra for College Students

Question

Explain how to write \(\sqrt{-64}\) as a multiple of \(i\)

Step-by-Step Solution

Answer

\(\sqrt{-64} = 8i\).

Step 1: Identify the square root of the negative number

The imaginary unit \(i\) is defined by its property \(i = \sqrt{-1}\). So, to express \(\sqrt{-64}\) as a multiple of \(i\), it's necessary to separate \(-64\) into \(-1*64\). Thus, \(\sqrt{-64} = \sqrt{-1*64}\).

Step 2: Apply the square root property

The square root of a product can be written as the product of the square roots if the numbers under the square root are both nonnegative. Since we have \(\sqrt{-1*64}\), we can rewrite it as \(\sqrt{-1}*\sqrt{64}\).

Step 3: Simplify the square roots

The square root of -1 is the imaginary unit \(i\), and the square root of 64 is 8. Therefore, \(\sqrt{-1*64}\) simplifies to \(8i\).