Question

The nth root of any positive real number has 2 real solutions if n is even, and 1 real solution if n is uneven.

However, I recently learned that the square root sign, is actually a principal square root, and denotes only the positive square root.

Is this standard symbol for roots generalizable for higher degree roots? Eg: does the fourth square root symbol also denotes the principal fourth square root? What about fractional exponents?

Answer 1

Depending on when you use it, the square root sign can mean either the principal square root or just square root, where your answer will involve a plus/minus sign.

Generally, the higher up you get in mathematics you will be expected to express all possible solutions to an equation, so use of the 'principal' square root is less common.

Answer 2

Try typing cube root of -8 into Wolfram.....

Answer 3

Isn't that wrong? And how is it relevant?

Answer 4

If you were asked to find:
\[\sqrt[4]{16}\]
When first learning about indices, your teachers would probably be happy for you to give the answer as 2.
However, as you learn more, you learn that there will be another solution, -2.

I understand what you're asking, the basic answer is that it depends if you've been taught about it yet. If you have, then you should be expected to give all possible roots (unless it says otherwise).

Answer 5

So for example, if you were given a question such as:
"The height of a flower (y) is found to be equal to the square root of the amount of plant feed (x) given to it (in g). Find the height of a flower which has been given 16g of plant feed."
So you would set up the equation:
\[y=\sqrt x\]
And sub in x=16 and solve for y to find that:
\[y=\sqrt 16 =\pm4\]
However, in the context of this question, a negative height would make no sense at all. So in this question dealing with height we would probably regard \sqrt as the principal square root.
Does this answer your question?

Answer 6

\[\sqrt{16} = 4\]
Because this is the symbol for principal square root. My question is about notation not about whether or not my teacher expects it or if it is relevant to the problem

Answer 7

Correct, and yes the fourth square root symbol does symbolise the principal fourth root. However as I was saying, as you move forward in mathematics it's very rare that you wouldn't automatically take both the positive and negative roots for completeness.