Recursive Exponents

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Tega1984: Senior | Next Rank: 100 Posts; Posts: 51; Joined: Wed Oct 13, 2010 4:47 am; Thanked: 2 times; GMAT Score:560

by Tega1984 » Tue Jun 14, 2011 2:32 am

Here's what I'm confused about. Let's say 2^2^3 and 2^(2)^3
These two expressions are different right? So in the first one we would simply take 2 to the power of 2 and get 4, and then take 4 to the power of 3 and get 64 right?
In the second one we would first take 2(the one that is in brackets) to the power of three, because of the order of the operations(we must start from brackets), get 8 and than take 2 to the power of 8 to get 256? Please correct me if I'm wrong.

Last edited by Tega1984 on Tue Jun 14, 2011 1:07 pm, edited 2 times in total.

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Source: Beat The GMAT — Problem Solving | Tue Jun 14, 2011 2:32 am

cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 amFollowed by: 123 members; Location: India; Thanked: 310 times; GMAT Score:750

by cans » Tue Jun 14, 2011 2:41 am

according to me, both are wrong
2^(2)^3 is same as 2^2^3. (we start from the brackets, but only 2 is in the bracket, so we don't know whether 2^2 is calculated first or 2^3.
brackets should be like 2^(2^3) (then we will solve 2^3 first)

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Tega1984: Senior | Next Rank: 100 Posts; Posts: 51; Joined: Wed Oct 13, 2010 4:47 am; Thanked: 2 times; GMAT Score:560

by Tega1984 » Tue Jun 14, 2011 3:43 am

Well what's wrong with the first one? We just take 2 to the power of 2 to get 4 and than cube 4 to get 64.

cans wrote:according to me, both are wrong
2^(2)^3 is same as 2^2^3. (we start from the brackets, but only 2 is in the bracket, so we don't know whether 2^2 is calculated first or 2^3.
brackets should be like 2^(2^3) (then we will solve 2^3 first)

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Tue Jun 14, 2011 12:50 pm

I think you may have written your brackets in the wrong place. There is an important difference between the following two expressions:

(2^2)^3 ---> this is equal to (2^2)(2^2)(2^2) = 2^6. This is the situation where the 'tower of powers' exponents rule may be used: (a^b)^c = a^(bc)

2^(2^3) ---> this is equal to 2^8, since we need to evaluate the expression in brackets first

When no brackets are included (which is rare, but I've seen that situation in a couple of official questions), then you work from the top down. So if you saw 2^2^3 with no brackets, that would be equal to 2^8. You won't often encounter that situation, however.

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Tega1984: Senior | Next Rank: 100 Posts; Posts: 51; Joined: Wed Oct 13, 2010 4:47 am; Thanked: 2 times; GMAT Score:560

by Tega1984 » Tue Jun 14, 2011 1:27 pm

Thanks very much Ian.Here's a question I got on the gmatprep test. It goes like this -->2^(4-1)^2/2^3-2 and this equals 2^(3)^2/2=2^9/2=2^8 and here only 3 is in brackets and not (3^2)

Ian Stewart wrote:I think you may have written your brackets in the wrong place. There is an important difference between the following two expressions:

(2^2)^3 ---> this is equal to (2^2)(2^2)(2^2) = 2^6. This is the situation where the 'tower of powers' exponents rule may be used: (a^b)^c = a^(bc)

2^(2^3) ---> this is equal to 2^8, since we need to evaluate the expression in brackets first

When no brackets are included (which is rare, but I've seen that situation in a couple of official questions), then you work from the top down. So if you saw 2^2^3 with no brackets, that would be equal to 2^8. You won't often encounter that situation, however.

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Tega1984: Senior | Next Rank: 100 Posts; Posts: 51; Joined: Wed Oct 13, 2010 4:47 am; Thanked: 2 times; GMAT Score:560

by Tega1984 » Tue Jun 14, 2011 1:47 pm

But now I realize that we don't need to use the brackets with 3....o.k. I got it now

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