Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(5-4 i)^{2}$$

Step-by-Step Solution

Answer

The result of the operation \((5-4i)^2\) in standard form is \(9 - 40i\).

Step 1: Write Down the Complex Number

First, write down the complex number. It is given as \( (5-4i) \). We need to square this complex number, so essentially, we have to multiply it by itself: \( (5-4i) * (5-4i) \).

Step 2: Apply FOIL Method

Now we will apply the FOIL method. Multiply the First terms together: \( 5 * 5 = 25 \). Multiply the Outer terms together: \( 5 * -4i = -20i \). Multiply the Inner terms together: \( -4i * 5 = -20i \). Multiply the Last terms together: \( -4i * -4i = 16i^2 \).

Step 3: Combine Like Terms

Next, we will add together the results of step 2. This will lead us to \( 25 - 20i - 20i + 16i^2 \).

Step 4: Simplify the Result

This step involves simplifying the result from step 3. Given that \( i^2 = -1 \), we substitute \( i^2 \) with -1 in the expression. So, we have \( 25 - 40i + 16*(-1) = 25 - 40i - 16 = 9 - 40i \).