Problem 29 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \sqrt{-64}-\sqrt{-25} $$

Step-by-Step Solution

Answer

The result in standard form is \(3i\)

Step 1: Understanding Imaginary Numbers

When we try to take a square root of a negative number, the result is an 'imaginary number'. This is because of the definition of the imaginary unit \(i\), which is defined as \(\sqrt{-1}\). Hence, any square root of a negative number can be represented as \(i \times\) the square root of the absolute value of that negative number. Thus we need to rewrite \(\sqrt{-64}\) and \(\sqrt{-25}\) using \(i\).

Step 2: Converting Square Roots

\(\sqrt{-64}\) can be rewritten as \(\sqrt{64} \times \sqrt{-1}\), which is \(8i\) and \(\sqrt{-25}\) can be rewritten as \(\sqrt{25} \times \sqrt{-1}\), which is \(5i\). Thus, the equation now becomes \(8i - 5i\).

Step 3: Performing the Operation

Now that we have simplified the terms, we can do the operation. Subtracting \(5i\) from \(8i\) we get, \(3i\). Hence, \(3i\) is the final result.