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If a and b are positive integers and (2^a)^b = 2^3, what is the value

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Bunuel
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If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   03 Jul 2016, 11:39
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If a and b are positive integers and \((2^a)^b = 2^3\), what is the value of \(2^a*2^b\)?

A) 6
B) 8
C) 16
D) 32
E) 64
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C
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   14 Apr 2017, 01:35
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Bunuel wrote:
If a and b are positive integers and (2^a)^b = 2^3, what is the value of 2^a*2^b?

A) 6
B) 8
C) 16
D) 32
E) 64


\({ \left( { 2 }^{ a } \right) }^{ b }={ 2 }^{ 3 }\\ { 2 }^{ ab }={ 2 }^{ 3 }\\ ab=3\)

ab must be \((1)(3)\) or \((3)(1)\)

\({ 2 }^{ a }\ast { 2 }^{ b }={ 2 }^{ a+b }\\ { 2 }^{ 1+3 }={ 2 }^{ 4 }=16\)
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If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   03 Jul 2016, 12:46
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Bunuel wrote:
If a and b are positive integers and (2^a)^b = 23, what is the value of 2^a*2^b?

A) 6
B) 8
C) 16
D) 32
E) 64


(2^a)^b = 2^3
=>2^ab = 2^3
ab = 3
if a = 1 then b = 3 or if a=3 then b=1

2^a*2^b = 2^(a+b)
= 2^4 = 16
Answer C
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   04 Jul 2016, 00:21
c-16

ab=3.Since a and b are positive integers the only factors of prime number(this is the gist) 3 are 3 and 1.

so a can be 3 or 1 and b can also be 3 or 1

2^a+b=2^1+3 or 2^3+1=2^4=16
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   04 Jul 2016, 09:13
(2^a)^b = 2^ab
=> ab =3
and a and b are both integral values,
=> 3 can only be expressed as 1 * 3 as 3 is prime number
therefore a =1 and b =3 or b =1 and a =3
2^a * 2^b = 2^(a+b) = 2^(4) = 16.
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   04 Jul 2016, 09:30
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ab=3. 3+1=4. Hence, the answer is 2^4= 16
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   08 Jul 2016, 11:44
Bunuel wrote:
If a and b are positive integers and (2^a)^b = 2^3, what is the value of 2^a*2^b?

A) 6
B) 8
C) 16
D) 32
E) 64


Simple solution. Pick values of a and b, such that the given equation is true. Here, a could be 1 and b could be 3. (2^1)^3 = 2^3. With these values of a & b, the expression (2^a)(2^b) evaluates to be (2^1)(2^3) = 2 x 8 = 16. Answer choice:
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C
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   01 Sep 2020, 09:53
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Bunuel wrote:
If a and b are positive integers and \((2^a)^b = 2^3\), what is the value of \(2^a*2^b\)?

A) 6
B) 8
C) 16
D) 32
E) 64


Given: \((2^a)^b = 2^3\)

Apply the Power of a Power rule to get: \(2^{ab} = 2^3\)

From this we can conclude that: \(ab = 3\)

Since a and b are positive integers, we can be certain that one of the values (a or b) is 1, and the other value is 3
This means the SUM of a and b is 4 (1 + 3 = 4)

We want to find the value of \(2^a*2^b\)

Apply the Product rule to get: \(2^{a+b}\)
Since we now know the SUM of a and b is 4, we can write: \(2^{a+b}=2^{4}=16\)

Answer: C

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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   01 Sep 2020, 10:35
Asked: If a and b are positive integers and \((2^a)^b = 2^3\), what is the value of \(2^a*2^b\)?

2^(ab) = 2^3
ab = 3
Since a & b are positive integers (a, b) = {(1,3),(3,1)}

a + b = 1 + 3 = 3+ 1 = 4
\(2^a*2*b = 2^{a+b} = 2^4 = 16\)

IMO C
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   02 Dec 2020, 08:58
Not sure if this is the PC way of doing this but this is how I cam to the right conclusion.

obviously we came to the conclusion that 2^ab=2^3 is equiv to ab=3. if ab=3 just plug that into 2^ab which is 2^3=16. Bingo
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   28 Feb 2023, 08:22
Bunuel wrote:
If a and b are positive integers and \((2^a)^b = 2^3\), what is the value of \(2^a*2^b\)?

A) 6
B) 8
C) 16
D) 32
E) 64

\((2^a)^b = 2^3\)

Or, \(2^{ab} = 2^3\)

So, \(ab = 3\)

Either \(a = 3\) and \(b = 1\) Or, \(a = 1\) and \(b = 3\) ; Further \(a + b = 4\)

Thus, \(2^a*2^b = 2^{(a+b)}\)

So, Thus, \(2^a*2^b = 2^4 = 16\), Answer must be (C)
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink] New post   01 Mar 2024, 06:16
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