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Question:

Given that x²⁰¹⁸y²⁰¹⁷ = 1/2 and x²⁰¹⁶y²⁰¹⁹ = 8, the value of x² + y³ is

Explanation:

x²⁰¹⁸y²⁰¹⁷ = 1/2 ...(1)

x²⁰¹⁶y²⁰¹⁹ = 8    ...(2)

Dividing the 2 given equations, we get

$\frac{x^{2018}{y}^{2017}}{x^{2016}{y}^{2019}}$ = $\frac{{\displaystyle \frac{1}{2}}}{4}=\frac{1}{16}$

x² = y²/16

From Eq (1)

x^2(1009)y²⁰¹⁷ = 1/2

∴ ${\left(\frac{y^2}{16}\right)}^{1009}{y}^{2017}$ = 1/2

∴ y⁴⁰³⁵ = 1/2 × 16¹⁰⁰⁹

∴ y⁴⁰³⁵ = 1/2 × 2^{4 × 1009} = 2⁴⁰³⁵

∴ y = 2

Again, from equation 1

x²⁰¹⁸y²⁰¹⁷ = 1/2

∴ x²⁰¹⁸2²⁰¹⁷ = 1/2

∴ x²⁰¹⁸ = 1/2²⁰¹⁸

∴ x = ± 1/2

Now, x² + y³ = 1/4 + 8 = 33/4

Hence, option (b).