I'm supposed to evaluate the following equation in the real number system f possible : (-9)^1/2.
The answer says it's not possible; why, I don't understand this.
The problem is really how to define fractional powers like x^(1/2).
You want it to define in such a way so that the usual rules for taking powers still hold. E.g.:
(x^a)^b = x^(ab)
This means that you want to define
x^(1/2) such that:
(x^(1/2))^2 = x
So, x^(1/2) should be the square root of x. But this is only defined for positive x. The square of a real number is always positive, so if x is negative, no real number can be its square root.
oh, okay, I see; thank you very much! :D
To evaluate the expression (-9)^(1/2), let's break it down step by step.
First, let's understand what the exponent 1/2 means. The exponent of 1/2 represents a square root operation. So, (-9)^(1/2) can be written as √(-9).
Now, in the real number system, we typically operate with positive numbers. When taking the square root of a positive number, we get a positive result. For example, √9 = 3, because 3^2 = 9.
However, when we try to take the square root of a negative number, like -9, we encounter a problem. In the real number system, there are no real numbers whose square is a negative number. In other words, √(-9) does not produce a real number.
This is why the expression (-9)^(1/2) is considered not possible in the real number system. It involves taking the square root of a negative number, which does not have a real solution.