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Can $(-25^{44})$ be rewritten as $(-1) (25)^{44}$?

Will the exponent 44 affect the (-1) because of the parentheses?

Or does the 44 need to be inside the parentheses: $(-1) (25^{44})$?

Thank you!

Alan
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GardenRoses
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    The better choice would be to write it as $-(25^{44})$ or as $(-25)^{44}$ depending on what is meant. I expect that both are possible interpretations of your original expression depending on context and region. If I had to pick one, I would expect it to be $-(25^{44})$ instead. – JMoravitz Aug 18 '21 at 16:31
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    @NoahSchweber you seem to have missed the point of the question. The question is whether the negative sign which signifies taking the additive inverse of the number is being applied to the $25$ before the exponentiation or if it is being applied to the result of the exponentiation. It is obvious that $(-1)^2\neq -(1^2)$ – JMoravitz Aug 18 '21 at 16:33
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    As for "does the 44 need to be inside the parentheses" $(25)^{44}$ is the same as $(25^{44})$... it doesn't matter. – JMoravitz Aug 18 '21 at 16:35
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    With (-1) (25)^44 - will 44 only affect (25), or can it also affect (-1)? – GardenRoses Aug 18 '21 at 16:41
  • $ab^n$ is interpreted as $a\cdot (b^n)$, the exponent of $44$ in the expression in your comment only affects the $25$ and not the $-1$. – JMoravitz Aug 18 '21 at 16:43
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    PEMDAS: https://en.wikipedia.org/wiki/Order_of_operations#Unary_minus_sign : unary minus is usually taken to have lower priority than exponentiation (indices), so $-25^{44}$ is usually taken to mean $-(25^{44})$ rather than $(-25)^{44}$. Wikipedia could not find a whole lot of examples (mainly in some programming languages/spreadsheets) where the opposite convention is accepted. –  Aug 18 '21 at 16:44
  • @Algebrology Where did $(-25)^{44}$ come from? Notice, the OP never used that expression anywhere. – JMoravitz Aug 18 '21 at 17:23
  • Whoever did the first edit changed the order of the symbols. I went and re-edited it to match the OP's actual question – Alan Aug 18 '21 at 17:44
  • @Alan: The only problematic edit I see has your name on it -- you removed the clarifying sentence that the OP himself had added to the question in revision 3. The only other non-OP edits (revisions 2 and 4) were pure formatting fixes that did not change any text, nor the order or any symbols. – Troposphere Aug 18 '21 at 17:59

3 Answers3

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From the OP's comments, I understand the question to be what the meaning of the notation $(-1)(25)^{44}$ should be, rather than a question about the meaning of $-25^{44}$.

With standard notation, powers bind tighter than multiplication. Think, for example, of the general form of a second degree polynomial: $$ ax^2 + bx + c $$ This means $a\cdot(x^2)+\cdots$ rather than $(a\cdot x)^2+\cdots$.

It works the same with $(-1)$ instead of $a$ and $(25)$ instead of $x$: $$ (-1)(25)^{44} \quad\text{means}\quad (-1)\cdot(25^{44})$$ (which is of course the same as $-25^{44}$).

Troposphere
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  • FYI someone edited the OP's question in a way that changed the meaning, not sure if you answered the original one or the edited one. It is fixed now. – Alan Aug 18 '21 at 17:46
  • @Alan: Actually I went primarily by the OP's clarifying comment (which he also at one point edited into the question itself, though that has now been lost in warring edits). – Troposphere Aug 18 '21 at 17:51
  • No worries, I didn't really bother to figure out who meant what when, I just alerted all the people who answered just in case when I saw the edit problem. – Alan Aug 18 '21 at 17:52
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These two are different

$\begin{align} (-25^{44})&= -25^{44}\\ (-25)^{44}&= 25^{44} \end{align}$

in the later case, $-1$ will be effected.

DSD
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  • Your first line is unhelpful. Of course $(x) = x$. If the negative sign was a confusion for the left hand side of the expression, it would remain a confusion for the right. Better to write $(-25^{44})=-(25^{44})$ – JMoravitz Aug 18 '21 at 17:25
  • FYI someone edited the OP's question in a way that changed the meaning, not sure if you answered the original one or the edited one. It is fixed now. – Alan Aug 18 '21 at 17:46
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It all depends on how you write it. The way you wrote it, $(-25^{44}$), how our order of operations go is first look for parenthesis, resolve what's in parenthesis....but since the parenthesis is around EVERYTHING, it is meaningless. IE

$$(blah)=blah$$ or in this case $$(-25^{44})=-25^{44}$$

In usual written mathematics, the $-$ in front of a number is taken as "Multiply by negative 1", so since exponents go first, it is indeed equivalent to

$$-(1)\cdot 25^{44}$$

Note that in SOME computer programming/scripting languages, the order is reversed, so context is important. https://en.wikipedia.org/wiki/Order_of_operations#Unary_minus_sign

In order for the $-$ to be part of the exponent, we would need the exponent to have been OUTSIDE the parenthesis

$$(-25)^{44}=(-1)^{44}\cdot 25^{44}=25^{44}$$

Alan
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