Solve each equation. (1/e)^-x = (1/e²)^x+1

Verified step by step guidance

1. Rewrite the bases on both sides of the equation to have the same base. Note that \(\frac{1}{e} = e^{-1}\), so rewrite the left side as \(\left(e^{-1}\right)^{-x}\) and the right side as \(\left(e^{-2}\right)^{x+1}\).
2. Apply the power of a power rule, which states that \(\left(a^m\right)^n = a^{m \cdot n}\), to simplify both sides: the left side becomes \(e^{-1 \cdot (-x)} = e^{x}\), and the right side becomes \(e^{-2 \cdot (x+1)} = e^{-2x - 2}\).
3. Since the bases are the same (both are \(e\)), set the exponents equal to each other: \(x = -2x - 2\).
4. Solve the resulting linear equation for \(x\) by first adding \$2x\( to both sides to get \)x + 2x = -2\(, which simplifies to \)3x = -2$.
5. Divide both sides by 3 to isolate \(x\), giving \(x = \frac{-2}{3}\).

Key Concepts

Properties of Exponents

Understanding how to manipulate exponents is essential, including rules like (a^m)^n = a^(mn) and a^(-m) = 1/a^m. These properties allow simplification and rewriting of expressions to solve equations involving powers.

Exponential Equations

An exponential equation involves variables in the exponent. Solving such equations often requires rewriting both sides with the same base or applying logarithms to isolate the variable.

Equality of Exponential Expressions

If two exponential expressions with the same positive base are equal, their exponents must be equal. This principle allows setting the exponents equal to each other to solve for the unknown variable.