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syc.sophia
Students
 
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Joined: Fri Apr 25, 2014 7:49 pm
 

Number Properties, Ch 5. p. 93

by syc.sophia Sun May 04, 2014 2:45 am

Hi,

I would like to ask how to solve the following, as I didn't understand the answer given in the book:

Check Your Skills
19. Can these expressions be simplified (i.e. reduced to a single term)?
c. 2(2^n + 3^n)

This is the answer in the book (p.96): "Half of this expression can be simplified, namely the part that involves the common base, 2: 2^n+1 + 2 x 3^n. It may not be much prettier, but at least you've joined up the common terms."

I don't understand why this is.
Thanks!!
Sophia
tommywallach
Manhattan Prep Staff
 
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Joined: Thu Mar 31, 2011 11:18 am
 

Re: Number Properties, Ch 5. p. 93

by tommywallach Mon May 05, 2014 1:38 pm

Hey Sophia,

I'll do my best here, though I'm not precisely sure what part of this you don't understand, so you'll have to let me know if I answer your question.

The rule with exponents is that they're almost impossible to simplify unless you have common bases. So when we factor this out, we get this:

2 * 2^n + 2 * 3^n

Another way to think of that is:

2^1 * 2^n + 2^1 * 3^n

If you know your exponent rules, you know that whenever you multiply two exponential expressions with a common base, you can add the exponents, so:

2^1 * 2^n = 2^(n+1)

However, there's no simplification you can do on:

2^1 * 3^n

If you were thinking "Isn't it just 6^n?", that is not the case. Just to prove it:

2^1 * 3^1 = 6^1

BUT

2^1 * 3^2 = 18 --> that's not 6^2
2^1 * 3^3 = 54 --> that's not 6^3
etc.

Does that answer your question?

-t