(x+1)^x+10=(x+1)^x+4

2 min read Jun 16, 2024

Solving the Equation (x+1)^(x+10) = (x+1)^(x+4)

This equation presents a unique challenge because it involves an unknown variable as both the base and the exponent. Let's break down how to solve it:

Understanding the Equation

Similar Bases: Notice that both sides of the equation have the same base: (x+1). This is crucial for our solution.
Different Exponents: The exponents are different: (x+10) on the left side and (x+4) on the right.

Solving the Equation

Equate the Exponents: Since the bases are the same, we can solve the equation by simply equating the exponents:

(x + 10) = (x + 4)
Simplify and Solve for x:

x + 10 = x + 4 10 = 4

This is a contradiction! There is no solution for x that satisfies the original equation.

Explanation

The contradiction arises because the only way for two powers with the same base to be equal is if their exponents are also equal. The equation (x + 10) = (x + 4) implies that 10 must equal 4, which is logically impossible.

Conclusion

The equation (x+1)^(x+10) = (x+1)^(x+4) has no real solutions. This type of equation highlights the importance of carefully considering the properties of exponents and bases when solving equations.