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

1.	Assuming that x, y, z are positive real numbers, simplify each of the following:(i) \((\sqrt{x}^{-3})^5\)(ii) \(\sqrt{x^3y^{-2}}\)(iii) \((x^{-2/3}y^{-1/2})^2\)(iii) \((\sqrt{x})^{\frac{-2}{3}}\sqrt{y}^4+\sqrt{x}{y}\frac{-1}{2}\)(v) \(\sqrt[5]{243}x^{10}y^{5}z^{10}\)(vi) \((\frac{{x}^{-4}}{{y}^{-10}})^{\frac{5}{4}}\)
Answer» (i) \((\frac{\sqrt1}{\sqrt3})^5\) = (1 / x^3/2) 5 = (1 / x ^3/2 ^{× 5}) = (1 / x ^15/2) (ii) \((\sqrt3^3/\sqrt{y}^2)\) = (x³ / y²) ^1/2 = x³ ^× ^1/2 / y² ^{× 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^-1/3y ² / (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

Assuming that x, y, z are positive real numbers, simplify each of the following:(i) \((\sqrt{x}^{-3})^5\)(ii) \(\sqrt{x^3y^{-2}}\)(iii) \((x^{-2/3}y^{-1/2})^2\)(iii) \((\sqrt{x})^{\frac{-2}{3}}\sqrt{y}^4+\sqrt{x}{y}\frac{-1}{2}\)(v) \(\sqrt[5]{243}x^{10}y^{5}z^{10}\)(vi) \((\frac{{x}^{-4}}{{y}^{-10}})^{\frac{5}{4}}\)

Answer»

(i) \((\frac{\sqrt1}{\sqrt3})^5\) = (1 / x^3/2) 5

= (1 / x ^3/2 ^{× 5}) = (1 / x ^15/2)

(ii) \((\sqrt3^3/\sqrt{y}^2)\) = (x³ / y²) ^1/2

= x³ ^× ^1/2 / y² ^{× 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^-1/3y ² / (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