Exponent Rule

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sam2304: Legendary Member; Posts: 1239; Joined: Tue Apr 26, 2011 6:25 amFollowed by: 26 members; Thanked: 233 times; GMAT Score:680

by sam2304 » Mon Jul 16, 2012 11:26 pm

I have a query regarding one of the exponent properties/rules.

(3^2)^4 = 3^8 => 8 is a result of 2 * 4. This is one of the exponent rules.

How to differentiate between these. How to apply the rule in the below case and which one is right.

(3^2)^(2^2) = (3^2)^4 = 3^8 => The value 8 comes as a result of multiplying 2 and 4.

(3^2)^(2^2) = (3^2)^4 = 3^16 => The value 16 comes as a result of 2 raised to the power of 4.

I am sure only one of these can be right. But I am missing something here. Would be glad if someone could elaborate more on this.

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Source: Beat The GMAT — Problem Solving | Mon Jul 16, 2012 11:26 pm

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Mon Jul 16, 2012 11:49 pm

sam2304 wrote:(3^2)^(2^2) = (3^2)^4 = 3^8 => The value 8 comes as a result of multiplying 2 and 4.

This is correct.

sam2304 wrote:(3^2)^(2^2) = (3^2)^4 = 3^16 => The value 16 comes as a result of 2 raised to the power of 4.

This is not.

Here is why.

(3^2) = (3*3)
(3^2)^4 = (3^2)*(3^2)*(3^2)*(3^2) = (3*3)*(3*3)*(3*3)*(3*3) = (3*3*3*3*3*3*3*3) = 3^8

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sam2304: Legendary Member; Posts: 1239; Joined: Tue Apr 26, 2011 6:25 amFollowed by: 26 members; Thanked: 233 times; GMAT Score:680

by sam2304 » Tue Jul 17, 2012 12:08 am

Anurag@Gurome wrote:
sam2304 wrote:(3^2)^(2^2) = (3^2)^4 = 3^16 => The value 16 comes as a result of 2 raised to the power of 4.
This is not.

Here is why.

(3^2) = (3*3)
(3^2)^4 = (3^2)*(3^2)*(3^2)*(3^2) = (3*3)*(3*3)*(3*3)*(3*3) = (3*3*3*3*3*3*3*3) = 3^8

Let's take another example
3^2^2^3 = 3^12 [12 = 2*2*3].
3^2^2^3 = 3^256

What I did in the second case was 3^2^2^3 = 3^2^(8) = 3^(256) and this is wrong because I exponentially raised the value of 2 by 3 and 2 by 8 in the next step instead of multiplying everything. So when it involves power raised to powers, we should just multiply all the powers leaving the base alone and shouldn't raise the powers exponentially, right ?

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Tue Jul 17, 2012 12:15 am

sam2304 wrote:Let's take another example
3^2^2^3

Without any brackets, 3^2^2^3 can be interpreted differently.
Here are few possibilities,

3^2^2^3 = (3^2)^(2^3) = 3^12
3^2^2^3 = (3^((2^2)^3) = 3^64
3^2^2^3 = (3^(2^(2^3)) = 3^256

But yes, the concept is ((a^b)^c)^d = a^(bcd)

confuse mind: Master | Next Rank: 500 Posts; Posts: 462; Joined: Wed Jan 19, 2011 1:08 pmFollowed by: 4 members; Thanked: 10 times

by confuse mind » Wed Jul 18, 2012 7:16 pm

what I do in general is - follow the brackets

first solve whatever it is inside any bracket and that the exponent of that number. (Since the exponent rule always confuses me)

If no brackets in exam then you have to go with the way it is written in the exam. if it is a ladder rising up, start solving at the topmost extreme.

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