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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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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink]
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]
(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]
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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]
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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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink]
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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: $$2^{a+b}=2^{4}=16$$

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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink]
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]
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]
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]
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Re: If a and b are positive integers and (2^a)^b = 2^3, what is the value [#permalink]
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