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Math Expert V
Joined: 02 Sep 2009
Posts: 55277
If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 67% (02:04) correct 33% (02:29) wrong based on 325 sessions

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If $$3^k + 3^k = (3^9)^{3^9}-3^k$$, then what is the value of k?

(A) 11/3
(B) 11/2
(C) 242
(D) 3^10
(E) 3^11 – 1

Kudos for a correct solution.

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CEO  D
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Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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9
5
Bunuel wrote:
If $$3^k + 3^k = (3^9)^{3^9}-3^k$$, then what is the value of k?

(A) 11/3
(B) 11/2
(C) 242
(D) 3^10
(E) 3^11 – 1

Kudos for a correct solution.

Answer: Option
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EMPOWERgmat Instructor V
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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2
1
Hi All,

The answer choices to this question are written in a way that helps us to avoid some of the math involved.

We're told that 3^K + 3^K = (3^9)^[3^9] - 3^K. We're asked for the value of K.

To start, we should move all like terms to one side...

3^K + 3^K + 3^K = (3^9)^[3^9]

The 'left side' can be rewritten....

3(3^K) = (3^9)^[3^9]

3^(K+1) = (3^9)^[3^9]

Both sides have the same "base 3", so since the "right side exponent" is clearly a BIG INTEGER, we know that...

K+1 = a BIG INTEGER
K = a BIG INTEGER - 1

There's only one answer that fits...

Final Answer:

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# Rich Cohen

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ Senior Manager  Status: Math is psycho-logical Joined: 07 Apr 2014 Posts: 413 Location: Netherlands GMAT Date: 02-11-2015 WE: Psychology and Counseling (Other) If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k? [#permalink] ### Show Tags 2 3^k+3^k=(3^9)^3^9 - 3^k --> moving the - 3^k to the other side: 3(3^k) = (3^9)^3^9 --> using the properties of powers, we are now adding the exponents: 3^k+1 = 3^ (3^2)^(3^9) (here we broke 9 into 3^2) k+1 = 3^2*3^9 k+1 = 3^11 k = 3^11 - 1. ANS E Attachments exponents.png [ 662.84 KiB | Viewed 5106 times ] Math Expert V Joined: 02 Sep 2009 Posts: 55277 If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k? [#permalink] ### Show Tags Bunuel wrote: If $$3^k + 3^k = (3^9)^{3^9}-3^k$$, then what is the value of k? (A) 11/3 (B) 11/2 (C) 242 (D) 3^10 (E) 3^11 – 1 Kudos for a correct solution. MANHATTAN GMAT OFFICIAL SOLUTION: The common term in this problem is the recurring base of 3. We will group like terms (i.e. all the terms with k on the left side, all the other powers of 3 on the right side), then simplify each power of 3 using exponent rules. $$3^k + 3^k = (3^9)^{3^9} - 3^k$$ $$3^k + 3^k + 3^k = (3^9)^{3^9}$$ $$3(3^k) = (3^9)^{3^9}$$ $$3^{(k + 1)} = 3^{(9*3^9)}$$ $$k + 1 = 9 * 3^9$$ $$k + 1 = 3^2 * 3^9$$ $$k = 3^{11} - 1$$ The correct answer is E. _________________ Current Student Joined: 22 Apr 2015 Posts: 45 Location: United States GMAT 1: 620 Q46 V27 GPA: 3.86 Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k? [#permalink] ### Show Tags Can anyone tell me what Im doing mathematically illegal. I start by seeing all the bases are the same so I go straight to the exponents. k+k=9x$$3^9$$-k 3k=$$3^2*3^9$$ 3k=$$(3^11)$$ k=$$(3^11)/3$$ k=$$3^10$$ I believe I'm doing something wrong in step 2 but not sure if it is or not can anyone confirm? Senior Manager  B Joined: 28 Jun 2015 Posts: 290 Concentration: Finance GPA: 3.5 Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k? [#permalink] ### Show Tags 3^k + 3^k = (3^9)^(3^9) - (3^k) 3(3^k) = (3^9)^(3^9) 3^(k+1) = 3^([3^2]*[3^9]) k+1 = 3^2 * 3^9 k+1 = 3^(2+9) = 3^11 so, k = (3^11 - 1). _________________ I used to think the brain was the most important organ. Then I thought, look what’s telling me that. EMPOWERgmat Instructor V Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 14209 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k? [#permalink] ### Show Tags 1 Hi xLUCAJx, The first error is right at the beginning - you cannot combine exponents in the way that you did: 3^K + 3^K is NOT 3^2K 3^K + 3^K = 2(3^K) When you add 3^K to both sides, the 'left side' becomes... 3^K + 3^K + 3^K This can be rewritten as... 3(3^K) = (3^1)(3^K) = 3^(K+1) Using these steps as your 'starting point', what would you do next? GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** # Rich Cohen Co-Founder & GMAT Assassin Follow Special Offer: Save$75 + GMAT Club Tests Free
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Manager  S
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Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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(3^9)^3^9= 3^(3^2*3^9)=3^11
now taking all k terms on one side
3^k+3^k+3^k = 3^(k+1)
comparing power as bases are same
we get k+1=3^11, k=3^11-1
E answer'
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If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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Anyone knows what is the maximum number of "aggregated exponents" GMAT is likely to test? I could only solve this question because it regarded only "2 aggregated powers":$$(3^9)^{3^9}$$. Does "3 aggregated powers" such as $${3^9}^{3^9}$$ are possible to appear on GMAT questions? How to solve them?
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Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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Bunuel wrote:
If $$3^k + 3^k = (3^9)^{3^9}-3^k$$, then what is the value of k?

(A) 11/3
(B) 11/2
(C) 242
(D) 3^10
(E) 3^11 – 1

Let’s simplify the given equation:

3^k + 3^k = (3^9)^(3^9) - 3^k

3^k + 3^k + 3^k = 3^(9 * 3^9)

Pull out the common factor 3^k from each term on the left side of the equation:

3^k * (1 + 1 + 1) = 3^(3^2 * 3^9)

3^k * (3) = 3^(3^11)

Note that the left side is now 3^k * 3^1, so we combine (add) the exponents:

3^(k + 1) = 3^(3^11)

k + 1 = 3^11

k = 3^11 - 1

Answer: E
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Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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Can anyone please explain how 3^3k is transformed to 3^k+1
Math Expert V
Joined: 02 Sep 2009
Posts: 55277
Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?  [#permalink]

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danz1ka19 wrote:
Can anyone please explain how 3^3k is transformed to 3^k+1

I think you mean how is $$3*3^k=3^{k+1}$$

Operations involving the same bases:
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
$$a^n*a^m=a^{n+m}$$

$$\frac{a^n}{a^m}=a^{n-m}$$

So, $$3*3^k=3^1*3^k=3^{k+1}$$

8. Exponents and Roots of Numbers

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_________________ Re: If 3^k + 3^k = (3^9)^3^9 – 3^k, then what is the value of k?   [#permalink] 07 Apr 2019, 07:17
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