Assuming that x, y, z are positive real numbers, simplify each of the following:

(i) (ii) (iii) (x^-2/3y^-1/2)² (iv) (v) (vi)

(i) ( )⁵ = (1 / x^3/2) ⁵

= (1 / x ^{3/2 × 5}) = (1 / x ^15/2)

(ii) (³ / y²) = (x³ / y²) ^1/2

= x^{3 × 1/2} / y^{2 × 1/2}

= x^3/2 / y

(iii) 1 / (x^2/3 y^1/2)²

= 1 / (x^{2/3 × 2} y^{1/2 × 2})

= 1 / x^4/3 y

(iv) (x^1/2)^-2/3 (y)² / (xy^-1/2)^1/2

= x^-1/3y² / (x^1/2y^-1/2×1/2)

= (x^-5/6) (y^9/4)

= (y^9/4) / (x^5/6)

(v) (243x¹⁰ y⁵ z¹⁰) ^1/5

= (3⁵)^1/5 x²yz²

= 3x²yz²

(vi) (y¹⁰ / x⁴)^5/4

= y ^{10 × 5/4} / x^{4 × 5/4}

= y^25/2 / x⁵