Answer:
The conjugate is 2+5i and the product is 29.
Step-by-step explanation:
The conjugate of a complex number [tex]a+bi[/tex] is [tex]a-bi[/tex]
Our complex number is 2-5i.
Here, a = 2 and b = -5
Thus, the conjugate of 2-5i is
[tex]2-(-5i)=2+5i[/tex]
Using this rule:
[tex](a+b)(a-b)=a^{2}-b^{2}[/tex]
And the fact that [tex]i^{2} = -1[/tex]
We can find the product of any complex number and its conjugate:
[tex](a+bi)(a-bi)=a^{2} - (bi)^{2}=a^{2} - b^{2} i^{2} = a^{2} + b^{2}[/tex]
As our complex number is [tex]2-5i[/tex], the product with its conjugate will be
[tex]2^{2} + (-5)^{2} =4+25=29[/tex]
