Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(6+7 i)^{2}$$

Step-by-Step Solution

Answer

The result of squaring the given complex number \(6 + 7i\) is \(-13 + 84i\).

Step 1: Write out the operation

Express the given operation as the multiplication of the complex number \(6+7i\) by itself, like this: \((6+7i) * (6+7i)\)

Step 2: Apply the distributive law

Next, apply the distributive law and multiply the terms from one bracket with every term in the second bracket, namely: \(6*6, 6*7i, 7i*6, 7i*7i\)

Step 3: Perform the Multiplication

Carry out the multiplication to get the four terms from the previous step: \(36, 42i, 42i, -49\) because \(i^2\) equals -1 in the last term \(7i*7i\).

Step 4: Combine Like Terms

Next, combine the like terms, the real numbers and the imaginary numbers separately. This leads to: \(36 - 49 + 42i + 42i\) = \(-13 + 84i\)