$\displaystyle -7^(3/4)$
is it $\displaystyle -343 ^ (1/4)$ or - $\displaystyle (343 ^ 1/4)$ or both?
isn't $\displaystyle -343 ^ (1/4)$ impossible because there can't be a positive root of a negative number?
Work really hard at improving your notation. Find a consistent syntax.
Generally, the this "^" is for indicating an exponent. After that, use parentheses liberally to clarify.
-7 to the 3/4 power is -7^(3/4) or (-7)^(3/4)
1/4 of -7 cubed is (1/4)*(-7)^3 or (-7^3)/4 or ((-7)^3)/4
It is time for you to expand your understanding of the Order of Operations.
If there is confusion, Exponents go before division, so 7^3/4 would be (7^3)/4.
What you may not know is how to treat "Unary Minus". Trust me on this. That is what it is called. Unary Minus is almost always FIRST - before anything else. Thus, -7^3 would almost never be -(7^3), it would be (-7)^3. There are exceptions. If you are writing a program, you MUST know how your code will be interpreted by your machine. I code in a language that is ALWAYS interpreted left to right unless parentheses indicate otherwise. However, even in this language, its Unary Minus is an exception and goes first.
Now what do you think?
As pickslides wrote,
$\displaystyle -7^{1/4} = -343^{1/4}$
But I don't know why he said, "This does not have a solution." Both evaluate to about -4.304.
There is a difference between
$\displaystyle -343^{1/4}$ and $\displaystyle (-343)^{1/4}$
(I assume these are what you were trying to write), and it's all order of operations.
For $\displaystyle -343^{1/4}$, you raise 343 to the 1/4 power, and then tack on the negative. You can also write this as $\displaystyle -\sqrt[4]{343}$.
For $\displaystyle (-343)^{1/4}$, you raise (-343) to the 1/4 power, but that gives you a non-real answer. You can also write this as $\displaystyle \sqrt[4]{-343}$.
See the difference?
From Wikipedia on Order of Operations:
"Unfortunately, there exist differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression −3^2 is interpreted to mean −(3^2) = −9, but in some applications and programming languages, notably the application Microsoft Office Excel and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus (negation) has higher precedence than exponentiation, so in those languages −32 will be interpreted as (−3)2 = 9. In any case where there is a possibility that the notation might be misinterpreted, it is advisable to use parentheses to clarify which interpretation is intended."
So, according to this, unary minus only has the highest precedence if it's in excel, bc, or your programming language... Since I'm using printed, not programmed, mathematics, the unary minus doesn't get precedence?