Mathematics

Problem

Let

t

be a positive real number satisfying

(t^{2})^{3}=64

. (i) Rewrite the given equation in the simpler form

t^{n}=b

, clearly stating the values of

n

and

b

. (ii) Solve the equation to find all real values of

t

. (iii) Prove that each value of

t

found in part (ii) is equal to the principal sixth root of 64. (iv) Without using a calculator, simplify

\sqrt[6]{64}

and verify that it matches the value of

t

obtained.