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655-705 (Hard)|   Exponents|      
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Bunuel
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What is the value of 4^3^2^1^2^3^4 Updated on: 02 Nov 2022, 20:10
Originally posted by Bunuel on 01 Sep 2022, 01:38.
Last edited by chetan2u on 02 Nov 2022, 20:10, edited 1 time in total.
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C
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65% (hard)

Question Stats:

53% (01:28) correct 47% (01:24) wrong based on 336 sessions

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What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148

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What is the value of 4^3^2^1^2^3^4 21 Feb 2023, 11:59
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Bunuel
What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148

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Official Solution:

What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. \(2,144\)
B. \(262,142\)
C. \(262,144\)
D. \(262,146\)
E. \(262,148\)


To evaluate \(4^{3^{2^{1^{2^{3^4}}}}}\), we need to remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example, \(a^{m^n}\) means we first compute \(m^n\) and then use that result as the exponent for \(a\). Therefore, \(a^{m^n} = a^{(m^n)}\). On the other hand, if we have \((a^m)^n\), we compute \(a^m\) first and then raise the result to the power of \(n\), so \((a^m)^n=a^{mn}\).

Let's evaluate the exponents of 4 in our expression. Since \(1^{2^{3^4}}\) equals 1, we can ignore it completely, which means we will be left with \(4^{3^2}\).

Using the exponent rule, \(4^{3^2} = 4^9\).

Since the GMAT is a timed test, we may not have the luxury of actually computing the value of \(4^9\). Instead, we need to use some shortcuts to arrive at the answer. Fortunately, we can notice that the units digit of each answer choice is different, and if we can determine the units digit of \(4^9\), we can identify the correct answer.

The units digit of \(4^k\), where \(k\) is a positive integer, alternates between 4 and 6 for odd and even values of \(k\), respectively. Since 9 is an odd number, the units digit of \(4^9\) is 4.

We can now eliminate any answer choices that do not end in 4. The only answer choice that ends in 4 is C, so our answer is \(4^{3^{2^{1^{2^{3^4}}}}} = 262,144\).


Answer: C
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What is the value of 4^3^2^1^2^3^4 01 Sep 2022, 01:44
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1 raise to power anything is 1.so, essentially it boil downs to 4^(3^2) =4^9=64^3. Now you don't have to calculate elimination works pretty well. The unit digit should be 4 and (a) can't be answer as it is very small.we are left with only (c)

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What is the value of 4^3^2^1^2^3^4 01 Sep 2022, 01:46
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Bunuel
What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148

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There is a 2-step approach that can be used here.

Step 1: 1 raised to any number is always going to be one. On the basis of this, we can simply ignore all the powers of 1. After this step, our question becomes what is \(4^{3^{2}}\)? When simplified further, the question asks us what is \(4^{9}\)?

Step 2: You can solve it but there is an easier method. Use cyclicity of numbers. It gives you this - 4,6,4,6,4,6 and so on. So the units digit of \(4^{9}\) will be 4. We are now left with options A and C.

\(4^{9}\) can be also written as \(16^{4} * 4\) and \(16^{4}\) is definitely more than 2144.


Answer C
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What is the value of 4^3^2^1^2^3^4 01 Sep 2022, 02:01
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Bunuel
What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148
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Answer: Option C

Step-by-Step Video solution by GMATinsight

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What is the value of 4^3^2^1^2^3^4 02 Nov 2022, 20:22
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Bunuel
What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148

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Two ways

1) Number properties
No calculations required in this.
The base is 4, and the cyclicity when raised to successive powers is 4,6,4,6….
Only A and C left.
A is very less as compared to the number.
If you know powers \(2^{10}=1024\). Thus answer should be C.

2) Calculations
\(4^{3^{2^{1^{2^{3^4}}}}}\)
Anything raised to power of 1 results in 1
\(4^{3^{2^1}} =4^{3^{2}}=4^9\)
Should finish with 4 or 6 because of cyclicity.
A can be discarded because it is very small.
D can be discarded as its last two digits are 46, which is not divisible by 4. ( Rule of divisibility by 4)
Only C remains.
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What is the value of 4^3^2^1^2^3^4 16 Jun 2025, 13:29
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Hi Bunuel. Just wanted to notify that Option A also ends in a 4, but, we can eliminate option A considering 4^6> 2144. Right?
Bunuel
Bunuel
What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148

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Official Solution:

What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. \(2,144\)
B. \(262,142\)
C. \(262,144\)
D. \(262,146\)
E. \(262,148\)


To evaluate \(4^{3^{2^{1^{2^{3^4}}}}}\), we need to remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example, \(a^{m^n}\) means we first compute \(m^n\) and then use that result as the exponent for \(a\). Therefore, \(a^{m^n} = a^{(m^n)}\). On the other hand, if we have \((a^m)^n\), we compute \(a^m\) first and then raise the result to the power of \(n\), so \((a^m)^n=a^{mn}\).

Let's evaluate the exponents of 4 in our expression. Since \(1^{2^{3^4}}\) equals 1, we can ignore it completely, which means we will be left with \(4^{3^2}\).

Using the exponent rule, \(4^{3^2} = 4^9\).

Since the GMAT is a timed test, we may not have the luxury of actually computing the value of \(4^9\). Instead, we need to use some shortcuts to arrive at the answer. Fortunately, we can notice that the units digit of each answer choice is different, and if we can determine the units digit of \(4^9\), we can identify the correct answer.

The units digit of \(4^k\), where \(k\) is a positive integer, alternates between 4 and 6 for odd and even values of \(k\), respectively. Since 9 is an odd number, the units digit of \(4^9\) is 4.

We can now eliminate any answer choices that do not end in 4. The only answer choice that ends in 4 is C, so our answer is \(4^{3^{2^{1^{2^{3^4}}}}} = 262,144\).


Answer: C
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What is the value of 4^3^2^1^2^3^4 16 Jun 2025, 13:45
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adityaprateek15
Hi Bunuel. Just wanted to notify that Option A also ends in a 4, but, we can eliminate option A considering 4^6> 2144. Right?
Bunuel
Bunuel
What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. 2,144
B. 262,142
C. 262,144
D. 262,146
E. 262,148

Fresh GMAT Club Tests' Question

Official Solution:

What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. \(2,144\)
B. \(262,142\)
C. \(262,144\)
D. \(262,146\)
E. \(262,148\)


To evaluate \(4^{3^{2^{1^{2^{3^4}}}}}\), we need to remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example, \(a^{m^n}\) means we first compute \(m^n\) and then use that result as the exponent for \(a\). Therefore, \(a^{m^n} = a^{(m^n)}\). On the other hand, if we have \((a^m)^n\), we compute \(a^m\) first and then raise the result to the power of \(n\), so \((a^m)^n=a^{mn}\).

Let's evaluate the exponents of 4 in our expression. Since \(1^{2^{3^4}}\) equals 1, we can ignore it completely, which means we will be left with \(4^{3^2}\).

Using the exponent rule, \(4^{3^2} = 4^9\).

Since the GMAT is a timed test, we may not have the luxury of actually computing the value of \(4^9\). Instead, we need to use some shortcuts to arrive at the answer. Fortunately, we can notice that the units digit of each answer choice is different, and if we can determine the units digit of \(4^9\), we can identify the correct answer.

The units digit of \(4^k\), where \(k\) is a positive integer, alternates between 4 and 6 for odd and even values of \(k\), respectively. Since 9 is an odd number, the units digit of \(4^9\) is 4.

We can now eliminate any answer choices that do not end in 4. The only answer choice that ends in 4 is C, so our answer is \(4^{3^{2^{1^{2^{3^4}}}}} = 262,144\).


Answer: C
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You are right, however A is out because 4^9 = 2^18 is for sure greater than 2,144, because already 2^11 = 2,048.
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What is the value of 4^3^2^1^2^3^4 16 Jun 2025, 13:57
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Thank you! Always grateful to you for your guidance and support.
Bunuel
adityaprateek15
Hi Bunuel. Just wanted to notify that Option A also ends in a 4, but, we can eliminate option A considering 4^6> 2144. Right?
You are right, however A is out because 4^9 = 2^18 is for sure greater than 2,144, because already 2^11 = 2,048.
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