No, it is really a reciprocal.

If two numbers are reciprocals of one another, they multiply to 1.

We must multiply the two conjugates and show that they are equal to 1.

362 x 362 = 131044.
209 x 209 x 3 = 131043.

Their difference is 1.

In fact, we do not even need to use the power 5 version to show that the terms are indeed reciprocal.

2 + sq 3 is the reciprocal of 2 - sq 3 simply because they multiply to 1.

If not for the fact that the question wants the candidate to use the power 5 version, I would rather use the power 1 version.

Oh wow, they actually multiply to 1? I dun goof'd for not checking haha. Good catch. Now wondering if I should delete my answer.

No need to delete the answer since you have put in the hard work and time to write the answer which is correct already; the reader should be able to decipher what I have supplemented from this comments section.

① Rationalising denominator method :

1/(2 + √3)^5

= 1/(362 + 209√3)

= 1/(362 + 209√3) x (362 - 209√3)/(362 - 209√3)

= (362 - 209√3)/(362² - (209√3)²)

= (362 - 209√3)/(131044 - 131043)

= (362 - 209√3)/1

= 362 - 209√3

= (2 - √3)^5 (shown)


OR


1/(2 + √3)^5

= (1/(2 + √3))^5

= ( 1/(2 + √3) x (2 - √3)/(2 - √3) )^5

= ( (2 - √3)/(2² - (√3)²) )^5

= ( (2 - √3)/(4 - 3) )^5

= ( (2 - √3)/1)^5

= (2 - √3)^5

(Shown)





② Direct multiplication method :

(2 + √3)^5 x (2 - √3)^5

= (362 - 209√3)(362 + 209√3)
= 362² - (209√3)²
= 131044 - 131043
= 1

Since (2 + √3)^5 x (2 - √3)^5 = 1,
then 1/(2 + √3)^5 = (2 - √3)^5

(Shown)


OR


(2 + √3)(2 - √3)
= 2² - (√3)²
= 4 - 3
= 1

Since (2 + √3)(2 - √3) = 1,

1/(2 + √3) = (2 - √3)

( 1/(2 + √3) )^5 = (2 - √3)^5

1/(2 + √3)^5 (1^5 is just 1)
= (2 - √3)^5

(Shown)