Complex Numbers

To find a+ib=‌

√−5−12i+√5+12i

√−8−6i

and calculate 2a+b :
Calculate √−5−12i and √5+12i :
Since the real parts of √−5−12i and √5+12i are positive:
‌√−5−12i=√‌

13−5

2

+i√‌

13+5

2

=2−3i
‌√5+12i=√‌

13+5

2

+i√‌

13−5

2

=3+2i
Calculate √−8−6i :
Given that the real part is negative:
√−8−6i=−√‌

10−8

2

+i√‌

10+8

2

=−1+3i
Find a+ib:

Using the calculations from the previous steps:
a+ib=‌

(2−3i)+(3+2i)

−1+3i

=‌

5−i

−1+3i

Multiply the numerator and the denominator by the conjugate of the denominator:
=‌

(5−i)(−1−3i)

(−1+3i)(−1−3i)

Simplify:
‌=‌

−5+i−15i−3

1+9

=‌

−8−14i

10

‌=−‌

4

5

−‌

7i

5

Thus, a=−‌

4

5

and b=−‌

7

5

.
Calculate 2a+b :
2a+b=2(−‌

4

5

)+(−‌

7

5

)=−‌

8

5

−‌

7

5

=−3
Hence, the result is -3 .