As it has been given that [tex]x=4+5i[/tex], [tex]y=2-9i[/tex].
We need to find the value of the following:
(i) [tex]4x-y[/tex], substituting the value of 'x' and 'y' in the expression, we get:
[tex]4(4+5i)-(2-9i)=4\times 4+4\times 5i-2+9i=16+20i-2+9i[/tex]
[tex]=14+29i[/tex]
So, [tex]4x-y=14+29i[/tex]
(ii) [tex]-x+3y[/tex], substituting the value of 'x' and 'y' in the expression, we get:
[tex]-(4+5i)+3(2-9i)=-4-5i+3\times 2-3 \times 9i=-4-5i+6-27i[/tex]
[tex]=2-32i[/tex]
So, [tex]-x+3y=2-32i[/tex]
(iii) [tex]x \times y[/tex], we need to substitute the value of 'x' and 'y' in the expression, for this, we can use distributive property of multiplication that says,
[tex]a(b+c)=a \times b+ a \times c[/tex]
Using the distributive property of multiplication:
[tex](4+5i)\times(2-9i)=4 \times 2-4 \times 9i+5i \times 2-5i \times 9i[/tex]
[tex]=8-36i+10i-45i^2[/tex]
Now, we know that [tex]i \times i=i^2=-1[/tex]
We get, [tex]8-36i+10i-45 \times (-1)=8-26i+45[/tex]
[tex]=8+45-26i=53-26i[/tex]
Therefore, [tex]x \times y[/tex]=[tex]53-26i[/tex].
(iv) We have, [tex]2x \times y[/tex], we need to substitute the value of 'x' and 'y' in the expression, we get:
[tex]2(4+5i)\times (2-9i)[/tex]
Again, we can use distributive property of multiplication that says,
[tex]a(b+c)=a \times b+ a \times c[/tex]
So,
[tex]2(4+5i) \times (2-9i)=2\times 4+2 \times 5i\times(2-9i)[/tex]
[tex]=8+10i \times(2-9i)=8\times 2-8 \times 9i+10i \times 2-10i \times 9i[/tex]
[tex]=16-72i+20i-90i^2[/tex]
since, [tex]i^2=-1[/tex]
we get,
[tex]16-72i+20i-90 \times (-1)=16-52i+90[/tex]
[tex]=106-52i[/tex]
Therefore,
[tex]2x \times y=106-52i[/tex]
