SOLUTION: Why does a base number to the exponent of zero equals 1? When a number is multiplied zero times, shouldn't it be zero? Example: 10 (exponent zero) = 1.

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Question 1206763: Why does a base number to the exponent of zero equals 1?
When a number is multiplied zero times, shouldn't it be zero?
Example: 10 (exponent zero) = 1.

Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Use the ^ key to indicate exponents.
Something like 10^2 means "10 squared" aka
On the keyboard, you find this symbol by holding "shift" then pressing the "6".


If 10^0 = 0 was the case, then multiplying both sides by 10 would get us the following:
10^0 = 0
10*10^0 = 10*0
10^1*10^0 = 0
10^(1+0) = 0 ............ use rule a^b*a^c = a^(b+c)
10^1 = 0
10 = 0
We run into a problem.
The two sides don't agree on the same number, in which we consider the last equation to be false.
The last equation being false makes 10^0 = 0 false.

But if 10^0 = 1 was the case, then we don't have any issues.
10^0 = 1
10*10^0 = 10*1
10^1*10^0 = 10
10^(1+0) = 10
10^1 = 10
10 = 10
The two sides match up to form a true equation at the end.
The true equation at the end leads to a domino effect to make the first equation true.

When going from something like 10^2 to 10^3 we multiply by 10.
Going in reverse from 10^3 to 10^2 we divide by 10.
10^2 to 10^1 is also "divide by 10".
And so on.

Here's a chart of select values.
10^31000
10^2100
10^110
10^01
10^(-1)1/10 = 0.1
10^(-2)1/(10^2) = 1/100 = 0.01
10^(-3)1/(10^3) = 1/1000 = 0.001

Multiply by 10 to move up the chart.
Divide by 10 to move down the chart.


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Here is the more general approach using any base b, where b is nonzero.
b^0 = 1
b*b^0 = b*1
b^1*b^0 = b
b^(1+0) = b
b^1 = b
b = b

Once again, b is nonzero.
If b = 0 was the case, then weird things start to happen and that's a very lengthy discussion for another day (and another class).
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
Why does a base number to the exponent of zero equals 1?
When a number is multiplied zero times, shouldn't it be zero?

Example: 10 (exponent zero) = 1. 

Anything DIVIDED by ITSELF, equals 1.

 

SAME result for all other numbers, variables, etc., EXCEPT 0.
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