Perfect Square

Question

For which value of n below is a perfect square

(2^8)+(2^11)+(2^n)

subhashghosh · Answer

2^8(1 + 2^3) + 2^n

= 2^8 * (9 + (2)^n-8)

= 9 + (2)^n-8 = 25 then its a perfect square

=> (2)^n-8 = 16

=> (2)^n-8 = 2^4

=> n = 12

hussi9 · Answer

yup thats the answer

jamifahad · Answer

@subhashghosh can you please explain me this part?

Thanks.

hussi9 · Answer

(2^8)+(2^11)+(2^n)

=(2^(4*2))+(2. 2^4 .2^6 )+ (2^n)

To complete the square n ==12 i.e 6*2

(2^2)^2 + +(2. 2^4 .2^6 ) +  (2^6)^2

amit2k9 · Answer

wish answer options were given

would have done answer fitting. 2^8 means 9+2^(n-8) = perfect square meaning 2^(n-8) = perfect square 16,64 are possible values. n=12 fits.

subhashghosh · Answer

@jamifahad, consider this :

2^8 * (9 + (2)^n-8) is a product of 2^8 (perfect square) and (9 + (2)^n-8)

So we want (9 + (2)^n-8) to be a perfect square.

Now 2^any number is even while 9 is odd

So we want 9 + even # to be a perfect square, and that will be and odd #.

So check for instance 9 + powers of two which can be a perfect square.

After a few calculations by checking powers of 2 in increasing order, you can see that 2^4 = 16, and 16 + 9 = 25, a perfect square.

Please let me know if you have any other query.

jamifahad · Answer

@subhash got it thanks.

gurpreetsingh · Answer

(2^8)+(2^11)+(2^n)

must be equal to (a+b)^2 form

=>(2^8)+(2^11)+(2^n) = a^2 + b^2 + 2ab

2^11 looks like a 2ab form . if we consider 2^8 = a^2 => a = 2^4

=> b must be 2^11 / (2*2^4) => b = 6

also n = 2b = 12

Spidy001 · Answer

2^8+2^11+2^n

taking 2^8 common we have

(2^8)(1+2^3+2^(n-8))
= (2^8)(9+2^(n-8))

we know that 2^8 is a perfect square .

so 9+2^(n-8) has to be perfect square

minimum power of 2 that can be added to 9 to make it a perfect square is 4.( 9+16 = 25)

=> n -8 =4

Hence n=12.

stne · Answer

just a small query, we are assuming that 2^n > 2^8 , which enables us to take 2^8 as a common factor 
and we get 2^8 ( 2^ n-8 )

but what if 2^n = 4 then n = 2 in which case we cannot take 2^8 as a common factor .

any thoughts as to how we can easily assume 2^n > 2^8 ?

EvaJager · Answer

Use the formula for  (a+b)^2=a^2+2ab+b^2. 
Since  2^8+2^{11}=(2^4)^2+2*2^4*2^6 , we need an extra term, that of  (2^6)^2=2^{12}  to complete the expression to a perfect square,  (2^4+2^6)^2.

So,  n  should be  12 .

stne · Answer

Thank you eva I think this approach is more justified, like it better

premnath · Answer

Your approach is absolutely right, except some ambiguity. Let me correct it.

Where from did you get n=2b ? Also, b is not equal to 6. Correct expression is  b=2^6

As  2^n = b^2 
So,  2^n= (2^6)^2=2^{12} 
Therefore, n=12

Perfect Square

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards