Question 1: Assuming that is a positive real number and are rational numbers, show:

Answer:

(xi) If , show that

Given

Question 2: Assuming are positive real numbers, simplify the following:

Answer:

Question 3: Assuming are positive real numbers, show that:

Answer:

(ii) If , prove that , where are different.

Question 4:

Answer:

RHS

LHS

Therefore RHS = LHS. Hence proved.

Question 5:

(i) If are distinct positive primes such that , find

(ii) If are distinct positive primes such that , find

Answer:

(i) If are distinct positive primes such that , find

(ii) If are distinct positive primes such that , find

Therefore

Hence

Question 6: If , find . Then compute the value of

Answer:

Therefore

Hence =

Question 7:

(i) If , show that

(ii) Determine , if

(iii) If and , find the value of

Answer:

(i) If , show that

We know:

Therefore

Therefore

Hence Proved.

(ii) Determine , if

Therefore

(iii) If and , find the value of

Similarly,

Hence

Question 8:

(i) If and , then show that

(ii) If , show that

(iii) If , show that

Answer:

(i) If and , then show that

Let

Therefore , ,

Now,

(ii) If , show that

Let

Therefore , ,

(iii) If , show that

Let

Therefore , ,

Question 9: Solve the following equations:

Answer:

Let

Therefore if

If

Let , Therefore

Therefore if

If

Let

Therefore if

If

Question 10: Given , find (i) the integral value of (ii) the value of

Answer:

Therefore