代数计算疑问：为何$1 - [1 - e^{-0.5/0.25}] = e^{-6}$而非$-e^6$？

Hey there, let's work through this confusion step by step—you’re totally right to flag that equation because it doesn’t add up as written! Let’s break down the math and fix the mix-up.

First, let’s unpack your original expression #

You wrote: $1 - [1 - e^{-0.5/0.25}]$

First, calculate the exponent inside the brackets:
0.5 / 0.25 = 2, so the exponent simplifies to -2. Now the expression is:
$$1 - [1 - e^{-2}]$$

Applying basic algebra: removing parentheses with a negative sign #

This is where the key rule comes in: when you have a negative sign in front of parentheses, you flip the sign of every term inside when you remove them.

So expanding the expression:
$$1 - 1 + e^{-2}$$

The $1$ and $-1$ cancel out, leaving you with $e^{-2}$—not $e^{-6}$, and definitely not $-e^6$.

Why might someone claim the result is $e^{-6}$? #

The only way that equation makes sense is if there was a typo in the exponent. For example, if the original exponent was -1.5/0.25 instead of -0.5/0.25:
1.5 / 0.25 = 6, so the exponent becomes -6. Then expanding:
$$1 - [1 - e^{-6}] = 1 -1 + e^{-6} = e^{-6}$$
That’s probably where the $e^{-6}$ came from—likely a copy error in the numerator of the exponent!

Clearing up the $-e^6$ confusion #

You wondered if the result should be $-e^6$, which mixes up two very different things:

• $e^{-6}$ means $\frac{1}{e^6}$—the negative sign is in the exponent, which tells you to take the reciprocal of $e^6$. This is always a positive number.
• $-e^6$ means the negative of $e^6$—here the negative sign is a coefficient in front of the exponential term, making it a negative number.

These are not interchangeable! To avoid this mix-up, remember: negative exponents = reciprocals, negative coefficients = flipping the sign of the whole term.

Quick recap of key algebra rules to avoid this mistake #

• Parentheses with negative signs: $a - (b - c) = a - b + c$ (flip all signs inside)
• Negative exponents: $x^{-n} = \frac{1}{x^n}$ (for positive $x$) — never rewrite this as $-x^n$

内容的提问来源于stack exchange，提问作者Darlene T. Banks