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Unfortunately, the first property you list is not a property of the zero of the real numbers and the second is not quite correct.

Property (2) should be restated as, "any nonzero real number x raised to the zero power equals 1: if x is a real number that is not equal to zero, then x^0=1."

In regards to (1),

0^0 is called an indeterminate form because the end result of trying to calculate 0^0 can be any real number: the end result is not a specific real number. The technical explanation involves calculus.

0^1 means raise zero to the first power: 0^1=0.

1

Zero raised to the zero or a negative power is undefined. Using exponent laws,

0^0 = 0^(1-1) = 0^1/0^1 = 0/0

If a < 0, 0^a = 1/(0^(-a)) = 1/0

You can't divide by zero.

If you raise a to the 0 power, you get:

a^0 = a^(1-1) = a^1/a^1 = a/a = 1,

provided a is not 0, as in that case you get the undefined 0/0.

FWIW, the 0^0 debate has raged for a long time. In address of your first property, mathematical debate has at least narrowed the discussion to 0^0=1 or 0^0=undefined (I have never seen anyone argue that 0^0=0).

The rest of the question (and your second property) seems to be over-thinking an otherwise obvious answer based on simple arithmetic identities...

0^x=0 for x>0 (as mentioned x<0 is undefined)

try it:

0^3=0*0*0=0

0^2=0*0=0

0^1=0=0

x^1=x for non-zero x

try it:

3^1=3=3

2^1=2=2

1^1=1=1

0^1=0=0

So here's the convention that mathematicians use:

0^x=0 for all x besides 0, and 0^0=1. This means the function 0^x is not continuous, but the function x^0=1 for all x is continuous. One of them has to give. One reason this is a good choice (as well as 0!=1), is so that simple formulas for power series expansions hold for all values of x. For example,

e^x=sum{i=0 to infinity} x^i/i!.

e^0=0^0/0! should be 1. (All other terms are 0.)

The properties you provide in your questions are true, however it is important to remember what raising to a power is doing. the power says how many of the base number there are. therefore regardless of the power. If the base number is zero, the answer will always be zero. Besides a rule you did not mention is that any number raised to a power of one is equals the number being raised.

6 1/2

zero raised to zero is 0 as any number raised to 1 is the number itself

Whoa! back up the truck!!!

you have misunderstood what you read.. or were taught.. or.. more than likely.. just forgot some of the details:

1) Zero raised to zero.. is NON-defined!! which is different from being zero.

2) Any NON-Zero number raised to zero power is one. (see rule 1 for zero)

Oh.. hmmm.. just read the other answers.. hehe.. Jim already beat me to the punch.. oh well

How do you conclude zero raise to zero is zero.

Anything raised to zero = 1 is a law of indices.

Anything raised to one is the number itself.

So, according to me,

Zero raise to zero = 1

Zero raise to one = 0