Why is $-3^4 = -81$ and $(-3)^4 = 81$?

Question

How do you express $-3^4$ to get an answer of $-81$. And how is $(-3)^4$ expressed to get the answer of $81$?

Answer 1

The first one is $$-(3)(3)(3)(3) = -(81) = -81$$

and the second one is

$$ \begin{align} &\underbrace{(-3)(-3)}(-3)(-3) \\ &= (9)\underbrace{(-3)(-3)} \\ &= (9)\quad(9) \\ &= 81 \end{align} $$.

Does that help?

Answer 2

Remember the order of operations. Evaluation inside parentheses comes first, then exponents, then multiplication and division, then addition and subtraction.

Negation can be fit into this scheme in a few ways. One option is as a hidden multiplication by the opposite of $1,$ that is: $$-a=(-1\cdot a).$$ A less typical (and a bit more cumbersome) option is as a hidden subtraction from $0,$ that is: $$-a=(0-a).$$ Neither of these is ideal as a definition of negation--the first requires a notion of $-1$ separate from the more general definition; the second requires subtraction, which is usually defined rigorously using negation--but fortunately, that isn't important here. The upshot is this: negation must come after exponents in the order of operations, regardless of whether you think of negation as hidden multiplication or hidden subtraction.

So, for $(-3)^4,$ we begin with the negation in the parentheses, then take the fourth power, yielding $81$ (as the negatives "cancel out" in pairs). On the other hand, for $-3^4,$ we begin with the fourth power, yielding $81,$ and then negate, yielding $-81.$

Answer 3

There are specific rlues which are conventions for interpreting mathematical expressions. We need such conventions so that a mathematical expression can be interpreted by everyone the same way. One is that the power is only applyed to the previous expression enclosed by brackets. If there are no brackets only the previous expression is included in the power.

Actually $$-3^4=-((3)^4)=-(81)=-81$$

But $$ (-3)^4=(-1\cdot 3)^4=(-1)^4\cdot(3^4)=1\cdot 81=81$$

So it is actually not a mathematical theorem, but rather a convention how mathematical expressions with powers need to be interpreted, without filling the expression with millions of brackets :).

Answer 4

It is just the way we represent something in maths. You can relate to scripts of a dialect .I mean to say you are questioning the way of representation of an idea by interpreting it in way different from the accepted one. All i am trying to say is that if you were to express the mathematical statements in English then you would read $$-3^4$$ as minus of three to the power of four and $$(-3)^4$$ as minus three to the power of four that is to say $$-3^4$$ implies multiply 3 four times and then negate the value. On the other hand $$(-3)^4$$ means take -3 multiply it 4 times with itself.