Question 1206763
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Use the ^ key to indicate exponents.
Something like 10^2 means "10 squared" aka {{{10^2}}}
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.
<table border = "1" cellpadding = "5"><tr><td>10^3</td><td>1000</td></tr><tr><td>10^2</td><td>100</td></tr><tr><td>10^1</td><td>10</td></tr><tr><td>10^0</td><td>1</td></tr><tr><td>10^(-1)</td><td>1/10 = 0.1</td></tr><tr><td>10^(-2)</td><td>1/(10^2) = 1/100 = 0.01</td></tr><tr><td>10^(-3)</td><td>1/(10^3) = 1/1000 = 0.001</td></tr></table>
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).
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