What is the Exponent Rule?

mcheytan

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Jan 25, 2008
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IS there a rule for adding exponents with same base and same power such as:

2^5 + 2^5 + 3^5 + 3^5 + 3^5= what?

i was just wandering is there a way we can simplify this without calculator and without going through each number separately.....for example combine the 2^5 together = 2(2^5)=4^5.......???
Can we do same thing for the 3^5s???
 

tkhunny

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Apr 12, 2005
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There is no such thing. There may be useful opportunities, but it is only coincidental. It's called the Distributive Property of Multiplication over Addition - often just The Distributive Property.


\(\displaystyle 2^{5} + 2^{5} = 2^{5}(1 + 1) = 2^{5}*2 = 2^{6}\)

This seems to be to our advantage

BUT

\(\displaystyle 2^{5} + 2^{5} + 2^{5} = 2^{5}(1 + 1 + 1) = 2^{5}*3\)

This is NOT particularly to our advantage as an "exponent" rule, per se.

See how convenient it was in your examples that there were 2 with a 2-base and 3 with a 3-base? Very misleading.
 

mcheytan

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so we can use the same thing if we have other numbers as base, for example for the 3^5s and /or 4^8+4^8+4^8..as long as the base and the power are the same we can apply this (1+1+1.....) and then just multiply (in the case where it will be helpful as for the 2^5+2^5=2^6??
 

tkhunny

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You seem to have missed my point. The rule you should use is the Distributive Property. Learn to factor and to simplify. Do NOT invent a new rule just for factoring and simplifying exponents with identical bases and just the right number of terms. You WILL become confused. There are just too many things to remember and too many possible situations. You will be FAR better served to stay with the more fundamental concepts.
 

mcheytan

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Jan 25, 2008
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Ok I got it!....sorry, i just always try to find a harder way to solve something simple...and make it complicated!
 
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