Find y if x⁴ y⁴ = 16
The equation x⁴ y⁴ = 16 presents a fascinating challenge in algebra, requiring a blend of exponent rules and root operations to isolate the variable y. Think about it: at first glance, the equation might seem complex, but by breaking it down into manageable steps, we can uncover the relationship between x and y. This article will guide you through the process of solving for y, explain the underlying mathematical principles, and address common questions to deepen your understanding.
Steps to Solve for y
To solve x⁴ y⁴ = 16 for y, we begin by isolating y using algebraic manipulation. Here’s a step-by-step breakdown:
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Divide both sides by x⁴:
Start by dividing both sides of the equation by x⁴ to isolate y⁴.
$ y⁴ = \frac{16}{x⁴} $
This step assumes x ≠ 0, as division by zero is undefined. -
Take the fourth root of both sides:
To solve for y, take the fourth root of both sides of the equation. Remember that the fourth root of a number can be positive or negative:
$ y = \pm \sqrt[4]{\frac{16}{x⁴}} $
This introduces the ± symbol, reflecting the two possible real solutions for y Simple, but easy to overlook. Practical, not theoretical.. -
Simplify the expression:
Simplify the fourth root of the fraction. The fourth root of 16 is 2, and the fourth root of x⁴ is x (since x⁴ = (x)⁴).
$ y = \pm \frac{2}{x} $
This gives the final solution: y = ±2/x Simple, but easy to overlook. Surprisingly effective..
Scientific Explanation: Exponents and Roots
The equation x⁴ y⁴ = 16 relies on the properties of exponents and roots. Let
