Online Number Properties #5

by **relentlesspursuito700plus** Sat Jun 21, 2008 4:09 pm

Okay, I got this correct, but I am not sure if I understand MGMAT's solution. Maybe I got lucky?

Anyway, can someone show me how 2^(x+1)= 2(2^x)? I know this works out mathematically if I plug in, say, 2 for 3 for x. But I am not sure that this would be obvious to me on test day.

Maybe it's because (2^x)+(2^x) is equal ot 2(2^x), which is equal to (2^1)(2^x) and could be further simplified to 2^(1+x) since they both have 2 as the common base?

Maybe I answerd my own question, but some sort of a confirmation would be really helpful.

Too bad this doesn't jump out at me in less than 2 minutes.

alternative email account 00 (the at symbol) gmail (the dot symbol) com

by **jasonmk2000** Sat Jun 21, 2008 8:33 pm

2^(x+1)= 2(2^x)

The reason why this is true is based on the property of exponents.

Take for example (This works only for multiplication):
2^2 x 2^3

Since the bases are the same add the exponents = 2^5
You can see that 2^5 is also the same as 2 x 2 x 2 x 2 x 2.

In reverse, we can also write 2^5 as (2^3 x 2^2) or (2^1 x 2^4) -- just as long as the exponents equal 5.

Similarly, in 2(2^x) we have two numbers with the same base (2) and (2^x) being multiplied to each other. Therefore, we can say (2^1) x (2^x) equals 2^(x+1) by adding the exponents.

I hope this helps.

by **rfernandez** Fri Jun 27, 2008 2:07 am

Yep, jasonmk2000 has it. It's a matter of seeing 2 as 2^1 and then using exponent properties to simplify, namely that when multiplying two powers with the same base, you rewrite the base and add the exponents.