Problem 46 - Algebra and Trigonometry

Question

Perform the indicated operation(s) and write the result in standard form. $$ (8+9 i)(2-i)-(1-i)(1+i) $$

Step-by-Step Solution

Answer

The result in standard form is \(23+10i\).

Step 1: Expand by distributing

Perform the distributive property in both parentheses to expand.\n\n$$ (8+9i)(2-i)= (8*2 + 8*(-i) + 9i*2 +9i*(-i)) $$. \nSimilarly, \n\n$$ (1-i)(1+i)= (1*1 + 1*i - i*1 - i*i) $$.

Step 2: Simplify using properties of imaginary numbers

We remember that \(i^2=-1\), and use this to simplify.\n\n$$ (8+9i)(2-i)= (16 - 8i + 18i -9i^2)=(16 + 10i - 9i^2)$$. \nSimilarly, \n\n$$ (1-i)(1+i)= (1 +i -i - i^2)= (1-i^2) $$. Continuing, the simplification becomes\n\n$$ (8+9i)(2-i)= (16 + 10i - 9*(-1))= (16 + 10i +9) = (25 + 10i)$$ \n\n$$ (1-i)(1+i)= (1-(-1)) = 1+1=2 $$.

Step 3: Execute the Subtraction Command

Subtract the result from step 2 (right parenthesis) from step 2 (left parenthesis):\n\n$$ ((8+9i)(2-i)) - ((1-i)(1+i)) = (25+10i) - 2= (25 -2) + 10i= 23+10i $$

Step 4: Write in standard form

As the result is in standard form \(a+bi\), no further simplification is necessary.