To multiply the complex numbers [tex]\( z = 19 + i \)[/tex] and [tex]\( w = 4 + 10i \)[/tex], we use the distributive property or the FOIL method. Here is a detailed, step-by-step solution:
1. Multiply [tex]\( z \)[/tex] and [tex]\( w \)[/tex]:
To find [tex]\( z \cdot w \)[/tex], we multiply each part of [tex]\( z \)[/tex] with each part of [tex]\( w \)[/tex]:
[tex]\( (19 + i)(4 + 10i) \)[/tex].
2. Expand using the distributive property or FOIL:
[tex]\[ z \cdot w = 19 \cdot 4 + 19 \cdot 10i + i \cdot 4 + i \cdot 10i \][/tex]
This results in four separate terms:
[tex]\[ = (19 \cdot 4) + (19 \cdot 10i) + (i \cdot 4) + (i \cdot 10i) \][/tex]
3. Calculate each term individually:
- Real part multiplications:
[tex]\[ 19 \cdot 4 = 76 \][/tex]
- Imaginary part multiplications:
[tex]\[ 19 \cdot 10i = 190i \][/tex]
[tex]\[ i \cdot 4 = 4i \][/tex]
- Combining real and imaginary with [tex]\( i^2 \)[/tex]:
[tex]\[ i \cdot 10i = 10i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex],
[tex]\[ 10i^2 = 10(-1) = -10 \][/tex]
4. Combine all terms:
Collect the real parts and the imaginary parts together:
[tex]\[ 76 + 190i + 4i - 10 \][/tex]
Combine the real parts [tex]\( 76 \)[/tex] and [tex]\( -10 \)[/tex], and the imaginary parts [tex]\( 190i \)[/tex] and [tex]\( 4i \)[/tex]:
[tex]\[ 76 - 10 = 66 \][/tex]
[tex]\[ 190i + 4i = 194i \][/tex]
5. Final answer:
Hence, [tex]\( z \cdot w \)[/tex] can be written as:
[tex]\[ z \cdot w = 66 + 194i \][/tex]
Therefore, the product of [tex]\( z \)[/tex] and [tex]\( w \)[/tex] is [tex]\( 66 + 194i \)[/tex].
In summary:
- The first step is to apply the distributive property or FOIL.
- The product of [tex]\( z \cdot w \)[/tex] is [tex]\( 66 + 194i \)[/tex].
- The real part is [tex]\( 66 \)[/tex].
- The imaginary part is [tex]\( 194i \)[/tex].
