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

Question

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

diegocml · Accepted Answer

{ \left( { 2 }^{ a } ight) }^{ 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

Skywalker18 · Answer

(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

hsbinfy · Answer

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

prady80 · Answer

(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.

ruggerkaz · Answer

ab=3. 3+1=4. Hence, the answer is 2^4= 16

DavidFox · Answer

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:  C

BrentGMATPrepNow · Answer

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

RELATED VIDEO
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Kinshook · Answer

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

njp1 · Answer

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

Abhishek009 · Answer

(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)

bumpbot · Answer

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

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