Does the negative apply before or after an exponent

Solution 1:

Neither of you is right - or at least, you are both half right.. $$-8^0=-8^{1/2-1/2}=\frac{-8^{1/2}}{-8^{1/2}}$$ Now we compromise so that everyone gets an equal chance: $$\frac{-2\sqrt 2}{2\sqrt 2 i}={-1\over i}=i$$ So the answer is $i$. The imaginary unit. The enigma of the conundrums. The unspeakable number.

I have divulged this information at great risk to our agents in the field. Please use this remarkable knowledge with discretion, and always remember: use more parentheses.

Solution 2:

Exponents take precedence over multiplication. Your friend is wrong.

Solution 3:

In general, operations are done in this order:

1. Brackets
2. Exponents
3. Division and Multiplication
4. Addition and Subtraction

For example, $10 + 0 \times 5$ isn't read as "$10 + 0 = 10$, $10 \times 5 = 50$", it's read as "$0 \times 5 = 0$, $10 + 0 = 10$".

Your example is slightly trickier as you have to recognize what the minus sign really means. "$-n$" is essentially shorthand for $0-n$, and realizing that makes the answer clear.

Your equation is "really" $0-8^0$. Looking at the list from earlier, we see that exponents are done before subtraction, so that step comes next. $0-8^0$ becomes $0-1$, which we write as $-1$. You are correct.

Really, though, the question is less than ideal. Writing it as $-(8^0)$, as you say, would clear up a lot of problems. It's very easy to misread or misunderstand a problem and steps clearly were never taken to avoid that here.

Solution 4:

You are correct: without parenthesis around the (−8), the formula −8⁰ is equivalent to (−1) * (8⁰).

You trying to determine whether the equation, −8⁰ = x, like −2³ = x, will result in x being positive or negative. 1 − 2³ = −7 can be written as 1 + −2³ = −7 or 1 + (−1) * (2³) = 7. Clearly the negative addition of terms is separate from the result of the exponent.