High School Algebra 2

Question

The multiplicative inverse of a complex number z is 1/z where z≠0. Find the multiplicative​ inverse, or​ reciprocal, of the complex number 5−11i. Then use complex conjugates to simplify. Check your answer by multiplying it by the original number.

Answer 1

This question can be broken into 3 steps:

1.      Write the multiplicative inverse of the given complex number { in this case it is 1/(5-11i) }

2.     "Use complex conjugates to simplify": this means multiplying the numerator and denominator of 1/(5-11i) by the complex conjugate of the denominator, which is found by changing the sign of the imaginary component. { this complex conjugate is 5+11i, so the simplification for this is set up like so: (1/(5-11i)) *((5+11i)/(5+11i)) }

3.      Check the answer to part 2 by multiplying it by 5-11i, and it is correct if you get 1 as your answer. { you should have (5+11i)/146 as your answer to part 2, and you will indeed find that this answer multiplied by (5-11i) is equal to 1 }

You will need a good grasp on some foundational concepts such as the distributive property and order of operations; let me know if you need help with these :)

Answer 2

The complex number you are starting with is 5 - 11i, so the reciprocal would be 1/(5 - 11i).

If we rationalize the denominator by multiplying by the number "1" as a complex number...

1/(5 - 11i) * (5 + 11i)/(5 + 11i) = (5 + 11i)/(25 + 132) = (5 + 11i)/(157)

This would be the reciprocal.