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705-805 Level|   Exponents|      
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Bunuel
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If 3^27^x = 27^3^x , then x is equal to 10 Nov 2020, 04:00
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75% (hard)

Question Stats:

56% (01:53) correct 44% (01:58) wrong based on 220 sessions

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If \(3^{27^x} = 27^{3^x}\) , then x is equal to

A. -1

B. -1/2

C. 1/2

D. 1

E. 2


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C
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chetan2u
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GMAT Focus 1: 735 Q90 V89 DI81
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If 3^27^x = 27^3^x , then x is equal to 10 Nov 2020, 04:10
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Bunuel
If \(3^{27^x} = 27^{3^x}\) , then x is equal to

A. -1

B. -1/2

C. 1/2

D. 1

E. 2
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Get 27 in terms of 3..
\(3^{27^x} = 27^{3^x}\)..........\(3^{(3^3)^x} = (3^3)^{3^x}\)........\(3^{3^{3x}} = 3^{3*3^x}\)
Equating the power of 3, we get =>\(3^{3x} = 3^{x+1}\)
Equating the power of 3 once again, we get =>\({3x} = {x+1}....x=\frac{1}{2}\)

C

Substitute x and see what fits in.
\(3^{27^x} = 27^{3^x}\) , then x is equal to

A. -1.............\(3^{27^1} = 27^{3^1}\).......... \(3^{27} = 27^{3}=3^9\) ....NO

B. -1/2.............\(3^{27^{-1/2}} = 27^{3^{-1/2}}\)............. \(3^{\frac{1}{\sqrt{27}}} = 27^{3}=3^9\) ....NO

C. 1/2.............\(3^{27^{1/2}} = 27^{3^{1/2}}\)............ \(3^{3^\frac{3}{2}} = (3^3)^{\sqrt{3}}= (3^{3^\frac{3}{2}}......\) ....YES

D. 1.............\(3^{27^1} = 27^{3^1}\)........... \(3^{27} = 27^{3}=3^9\) ....NO

E. 2.............\(3^{27^2} = 27^{3^2}\)............. \(3^{3^6} = 27^{3}=3^9\) ....NO
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If 3^27^x = 27^3^x , then x is equal to 12 Nov 2020, 01:02
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Bunuel
If \(3^{27^x} = 27^{3^x}\) , then x is equal to

A. -1

B. -1/2

C. 1/2

D. 1

E. 2
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If 3^27^x = 27^3^x , then x is equal to 28 Nov 2021, 12:49
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We should have parentheses to define the order of these exponents. 3^(27^x) is not the same as (3^27)^x.
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If 3^27^x = 27^3^x , then x is equal to 29 Nov 2021, 01:15
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WheatyPie
We should have parentheses to define the order of these exponents. 3^(27^x) is not the same as (3^27)^x.
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\(a^{m^n} = a^{(m^n)}\) and not \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down).

On the other hand, \((a^m)^n=a^{mn}\).

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If 3^27^x = 27^3^x , then x is equal to 11 Feb 2025, 04:17
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Hi,

I don t understand what you do in the equiation. ¿Could you correct me?
1. (3)^(27)^(X) = (3)^( 3^3 )^(X) = (3)^ 3^(3X) = 3 ^ (9X)
2. (27)^(3)^(X) = (3^3) ^(3)^(X) = (3^3) ^(3X) = 3 ^ (9X)

9X = 9X. The only possible answer should be cero.
Bunuel
WheatyPie
We should have parentheses to define the order of these exponents. 3^(27^x) is not the same as (3^27)^x.

\(a^{m^n} = a^{(m^n)}\) and not \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down).

On the other hand, \((a^m)^n=a^{mn}\).

8. Exponents and Roots of Numbers



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ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread
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If 3^27^x = 27^3^x , then x is equal to 11 Feb 2025, 05:07
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MarceloY
Hi,

I don t understand what you do in the equiation. ¿Could you correct me?
1. (3)^(27)^(X) = (3)^( 3^3 )^(X) = (3)^ 3^(3X) = 3 ^ (9X)
2. (27)^(3)^(X) = (3^3) ^(3)^(X) = (3^3) ^(3X) = 3 ^ (9X)

9X = 9X. The only possible answer should be cero.
Show more
1. and 2. are simplified as below,

1. \(3^{(27^x)} = 3^{(3^{3x})}\)

2. \(27^{(3^x)} = 3^{(3*3^x)} = 3^{(3^{x+1})}\)

-----------------------------------------------------------------------------

\(3^{(3^{3x})} = 3^{(3^{x+1})}\)

\(3^{3x} = 3^{x+1}\)

\(3x = x+1\)

\(2x = 1\)

\(x = \frac{1}{2}\)
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