Hello forum, Vaironxxrd here.
I have a question about exponents. If I have,
$\displaystyle (-9)(-9)^3$ Would that be $\displaystyle (-9)^4$ or $\displaystyle (-9^4)$.
If possible can you guys provide other examples?
It's indeed $\displaystyle (-9)^4$. For example, consider $\displaystyle (-2)^4\cdot (-2)^2$ which is offcourse $\displaystyle (-2)^6=64$ and $\displaystyle -2^6=-(2^6)=-64$ therefore there're not equal.
So it's important to use brackets!
Exponents come before multiplication in the order of operations and a minus sign in front is multiplying by -1.
However, brackets come before exponents so if the multiplication is done inside the bracket it too is affected by the exponents.
$\displaystyle (-9)^2 = -9 \times -9 = 81 \text{ OR } (-9)^2 = (-1 \times 9)^2 = (-1)^2 \times 9^2 = 81$
$\displaystyle -9^2 = -1 \times 9^2 = -81$ - the brackets here are unnecessary and only serve to complicate matters IMO.
Remember BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction).
The order of operands is very important, and this is where you're having trouble.
-9 is essentially like saying 0 - 9, you just omit the 0.
Now, from BIDMAS, we know that everything in brackets is evaluated first, and that indices/multiplication is evaluated before subtraction.
So (-9^2) is like saying (0 - 9^2). You evaluate the index first, then you subtract the result from 0.
0 - (9)(9) = 0 - 81 = -81
Whereas, in (-9)^2 you evaluate everything in brackets first, so (0 - 9)^2 = (-9)(-9) = 81, because multiplying an even number of negatives together results in a positive.
I hope that was clear enough.