The first step in multiplying two complex numbers, [tex]\( z = 19 + i \)[/tex] and [tex]\( w = 4 + 10i \)[/tex], is to apply the distributive property or FOIL (First, Outer, Inner, Last). This allows us to systematically multiply every term in the first complex number by every term in the second complex number.
To visualize this, we start with:
[tex]\[ z \cdot w = (19 + i) \cdot (4 + 10i) \][/tex]
Applying the distributive property or FOIL:
1. First: Multiply the real parts:
[tex]\[ 19 \times 4 = 76 \][/tex]
2. Outer: Multiply the real part of [tex]\( z \)[/tex] by the imaginary part of [tex]\( w \)[/tex]:
[tex]\[ 19 \times 10i = 190i \][/tex]
3. Inner: Multiply the imaginary part of [tex]\( z \)[/tex] by the real part of [tex]\( w \)[/tex]:
[tex]\[ i \times 4 = 4i \][/tex]
4. Last: Multiply the imaginary parts:
[tex]\[ i \times 10i = 10i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex], the last part becomes:
[tex]\[ 10i^2 = 10(-1) = -10 \][/tex]
Now, combining all these results:
- The real parts are [tex]\( 76 \)[/tex] and [tex]\( -10 \)[/tex], which sum to [tex]\( 66 \)[/tex].
- The imaginary parts are [tex]\( 190i \)[/tex] and [tex]\( 4i \)[/tex], which sum to [tex]\( 194i \)[/tex].
Thus, the product [tex]\( zw \)[/tex] is:
[tex]\[ zw = (66) + (194i) \][/tex]
So, the first step in determining [tex]\( zw \)[/tex] is to apply the distributive property or FOIL. From the steps outlined above, we get the resulting real and imaginary parts to be 66 and 194, respectively.
