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(x+1 5)^8+(2 7-y)^10=0

2 min read Jun 16, 2024
(x+1 5)^8+(2 7-y)^10=0

Exploring the Equation (x+15)^8 + (27-y)^10 = 0

This equation presents an interesting problem in algebra. Let's break down its key features and discuss potential approaches to understanding it.

Understanding the Equation

  • Even Powers: The exponents 8 and 10 are both even. This means that both terms on the left side of the equation will always be non-negative, regardless of the values of x and y.
  • Sum of Squares: The equation can be interpreted as the sum of two squared terms: [(x+15)^4]^2 + [(27-y)^5]^2 = 0.
  • Zero Solution: The only way for the sum of two non-negative terms to equal zero is if both terms are individually equal to zero.

Finding Solutions

Based on our analysis, we can deduce the following:

  1. (x+15)^4 = 0
  2. (27-y)^5 = 0

Solving these equations leads us to:

  1. x + 15 = 0 => x = -15
  2. 27 - y = 0 => y = 27

Therefore, the only solution to the original equation is x = -15 and y = 27.

Key Points

  • The equation has a unique solution.
  • Understanding the properties of even powers and the concept of squares is crucial for solving this type of equation.
  • The solution process involves setting each term individually to zero and then solving for the variables.

This equation provides a good example of how understanding basic algebraic principles can lead to solving complex-looking problems.

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