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 July 14th, 2012, 08:42 AM #1 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Exponent rule question If you have: $2^{3^{4}}$ Is it the same as $2^{(3 \cdot 4)}$ or $(2^3)^{4}$ or $2^{(3^{4})}$?
 July 14th, 2012, 08:44 AM #2 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Exponent rule question I'm leaning towards the first one but I just want to be sure
 July 14th, 2012, 09:03 AM #3 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,301 Thanks: 1023 Re: Exponent rule question Try it on a simple calculator
 July 14th, 2012, 09:19 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Exponent rule question The first two are equivalent to $2^{12}$ but you want $2^{81}$, which is the third one.
 July 14th, 2012, 09:32 AM #5 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Exponent rule question So if I wanted to simplify $3^{(2x+1)^{(3x+2)}}$, it would not be possible? I cannot multiply exponents that are raised to exponents? Because I remember learning that: $(x^{2})^{4}= x^{8}$ But wanted to make sure it's not the same thing as what's being presented here
 July 14th, 2012, 09:36 AM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Exponent rule question You have to consider what the exponents are raising. In $$$x^2$$^4$ the base for the exponent 4 is $x^2$, but with $x^{2^4}$ the base for the exponent 4 is 2, and the base for the exponent $2^4$ is x.
 July 14th, 2012, 09:37 AM #7 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Exponent rule question Oh, I see it now. Thanks, just wanted to clarify a differential question
 July 14th, 2012, 12:19 PM #8 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Exponent rule question $(a^b)^c= \underbrace{a^b \cdot \: a^b \cdot \: a^b .... }_{\text{c times}} = \underbrace{\underbrace{a \cdot \: a \cdot \: a ....}_{\text{b times}} \cdot \: \underbrace{a \cdot \: a \cdot \: a ....}_{\text{b times}} \: .....}_{\text{\text{c times}}} = \underbrace{a \cdot \: a \cdot \: a ....}_{\text{\text{\text{bc time}}}} = a^{(bc)}$ $\text{But}$ $a^{(b^c)}= \underbrace{a \cdot \: a \cdot \: a ....}_{\text{b^c times}} \ne \underbrace{a \cdot \: a \cdot \: a ....}_{\text{bc times}} \text{ since } b^c \ne bc$
July 14th, 2012, 05:42 PM   #9
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Re: Exponent rule question

Quote:
 Originally Posted by daigo If you have: $2^{3^{4}}$ Is it the same as $2^{(3 \cdot 4)}$ or $(2^3)^{4}$ or $2^{(3^{4})}$?
If the problem were $(2^3)^4$, then it would be equal to $2^{12}= 4096$.

But what you wrote (according to the latex code) is $2^{(3^4)}= 2^{81}$ which is much larger (on the order of $10^{24}$).

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