Can someone please explain why (-9)1/2 is not a real number but -9 1/2 is a real number

This only makes sense if you meant

√(-9) vs - (√9)

remember we cannot take the square root of a negative number, that would be an imaginary or complex answer.
but by the order of operation for -√9
we take the square root first, and then make the answer negative, so
- √9 = -3

So are you saying that (-9)1/2 and

-9 1/2 is not a real number because that is how the problem was written

I assumed that you meant

(-9)^(1/2)
which is the same as √-9 , which of course is not real

vs

-(9^(1/2))
which is -√9
which is -3

the ^ is the generally accepted symbol to indicate an exponent,
e.g. x^2 is x squared, x^5 is x to the fifth, etc

Sure, I'd be happy to explain it to you. Think of it this way: when you take the square root of a negative number, it does not yield a real number. So, (-9)1/2 is not a real number because you can't take the square root of -9.

However, when you have -9 1/2, you're dealing with a combination of a negative number (-9) and a fraction (1/2). The negative sign in front represents that the value is negative, and the fraction represents a part of a whole number. In this case, -9 1/2 is a real number because it's just a negative number combined with a fraction.

So, in summary, while (-9)1/2 is not a real number due to the impossibility of taking the square root of a negative number, -9 1/2 is a real number because it represents a negative number combined with a proper fraction.

To understand why (-9)^(1/2) is not a real number while -9 1/2 is, let's break down the concept of exponentiation and represent the calculation step by step.

1. (-9)^(1/2):
In mathematics, raising a number to a fractional exponent involves finding a root. Specifically, raising a number to the 1/2 power means finding its square root. However, the square root of a negative number is not a real number. In our case, the square root of -9 is an imaginary number, denoted as "i" in complex number systems.

So, (-9)^(1/2) is not a real number because the square root of -9 is not a real number. It falls under the category of complex numbers.

2. -9 1/2:
On the other hand, -9 1/2 represents a negative number (-9) combined with a fractional part (1/2). This expression is read as negative nine and one-half. It is a combination of a whole number and a fraction.

This is a real number because it can be represented on the number line. In decimal form, -9 1/2 can be written as -9.5.

In summary, (-9)^(1/2) is not a real number because the square root of a negative number is imaginary. But -9 1/2 is a real number since it represents a negative whole number combined with a fractional part.