√u2 =u for any number u.?
the square root of U to the power of 2 = u for any number u. True or False.
one thing I'm a bit indecisive is that if u is a negative number let's say -2 then there will be two answers + and - 2. And so + and - 2 is not really equal to -2 because there's still that + 2. Well the question didn't say I can't just give them one answer to satisfy u which in this case is -2.
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Hi,
√u2 = ±u
for example : (-2)^2 = 4 and 2^ = 4
=> √4 = ±2
You made a mistake in your comment. If u = +2 or u = -2 the quantity u^2 is still +4.
The principal square root of +4 is +2. Read the rest of this to see why that is true.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A positive real number has two real square roots,
because, for example,
3 Ã 3 = 9 but also
-3 Ã -3 = 9
So the square roots of 9 are +3 and -3.
To avoid getting tied up in semantics, (when someone loosely refers to the square root of 9), mathematics requires a clear unambiguous approach, and so we distinguish between these two by saying that if x is a non-negative real number it has just one non-negative square root, and we call this the PRINCIPAL square root.
By convention the PRINCIPAL square root is denoted by a radical sign
as âx
For positive x, this principal square root can also be written in exponent notation,
as x^(1/2).
So the principal square root of 9 is written â9 or 9^(1/2) and is equal to +3 and not -3.
~~~~~~~~~~~~~~~~
Your question asks
Is the â(u^2) = u for any number u? True or False.
1) Well, if u is limited to being a REAL quantity, (positive or negative), then u^2 is positive.
Since the convention is that the radical sign refers only to the PRINCIPAL square root which is positive then for real u only, â(u^2) = +u and in this case only your reply would be TRUE.
2) Careful though, the tricky questioner did not specify REAL u but used the broader word "any".
If u is already the IMAGINARY quantity u = -i = -â(-1) what happens then ?
Well u^2 is now (-1)*(-1)* i^2 which yields the negative quantity -1.
Any complex number, (which includes totally real or totally imaginary quantities), has two square roots. The two square roots of -1 are +i and -i.
But we still have the convention in place that the radical sign refers only to the PRINCIPAL square root so, â(-1) is equal to +i and not -i.
So here we have found an exception. When u = -i
â(u^2) is not = u
Conclusion: FALSE
Following the radical sign for principal square convention, the rule
â(u^2) = u for ANY number u
does NOT work for negative imaginary quantities.
Regards - Ian