Exponents and Parentheses

Question

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!

Answer 1

These two are different

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

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

Answer 2

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 $-25^{44}$).

Answer 3

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

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