GRE problem with exponents

[jsn09c]

Please help me in understanding why my answer is incorrect according to my book.

The question and my way of answering it is attached in a photo. The books explanation is also attached. Please help.

 

[pka]

Can you tell us why [imath]a+2b=6~?[/imath]
If so, then which has to be greater, [imath]a\text{ or }b~?[/imath]

 

[BigBeachBanana]

Please help me in understanding why my answer is incorrect according to my book.

The question and my way of answering it is attached in a photo. The books explanation is also attached. Please help.

The point the solution is making is that there are many solutions that satisfy [imath]a+2b=6[/imath], and you only pointed out one of them.
[math](a,b)=\{(0,3),(6,0),(2,2),...\}[/math] With these examples, you can see that their sums are equal to 3 or greater than 3. Therefore, we can't say for certain that it's greater than 3.

 

[jsn09c]

Thank you.

 

[jsn09c]

Can you tell us why [imath]a+2b=6~?[/imath]
If so, then which has to be greater, [imath]a\text{ or }b~?[/imath]

This is because prior to this the equation is:
2^a + 2^2b = 2^6
Since all have the same base of 2 you can rewrite the equation as:
a + 2b = 6

 

[lookagain]

This is because prior to this the equation is:
2^a + 2^2b = 2^6
Since all have the same base of 2 you can rewrite the equation as:
a + 2b = 6

No, your first equation is incorrect, and the second equation does not correctly
follow from it. Look at this:

(2^a)(4^b) = 64 --->

(2^a)[(2^2)^b] = 2^6 --->

(2^a)[2^(2b)] = 2^6 --->

2^(a + 2b) = 2^6 --->

a + 2b = 6

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And, for example, in 2^(2b), you must have the exponent, 2b, inside of grouping
symbols, such as parentheses, brackets, or braces.