If 3^k + 3^k = (3^9)^3^9 3^k, then what is the value of k? : Problem Solving (PS)

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

13 Jun 2015, 16:58

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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:

Show Spoiler

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]

14 Jun 2015, 04:59

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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

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

02 Jul 2015, 07:55

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?

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

02 Jul 2015, 12:04

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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?

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

18 Feb 2016, 04:32

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]

02 Feb 2017, 12:12

Expert Reply

giddi77 wrote:

If 3^k + 3^k = (3^9)^(3^9) - 3^k, then k = ?

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

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

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

Note that on the left side, when we add 3^k to itself three times, we have 3(3^k).

3(3^k) = (3^9)^(3^9)

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

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

Since the bases are equal, we can equate the exponents:

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]

02 Mar 2017, 18:09

Expert Reply

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]

06 Nov 2018, 04:18

Hi ,
I was able to solve it but took about 7 mints

.. my initial approach was :
3^k+3^k+3^k = (3^9)^(3^9)
3(3)^k =3(3^8)^(3^9)
3^k=(3^8)^(3^9)
k=8*(3^9)
I wasnt able to proceed beyond this and hence went back and re worked on it using the approach given by Brunnel. But out of curiousity, can anyone help me with where I went wrong in my approach?

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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, 07:10

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

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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

Expert Reply

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

Theory
Exponents and Roots on the GMAT: Tips and hints
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Questions
Units digits, exponents, remainders problems
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Tough and tricky PS exponents and roots questions with detailed solutions
Exponents DS Questions
Exponents PS Questions
Roots DS Questions
Roots PS Questions

Check below for more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

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

30 Aug 2019, 05:36

ScottTargetTestPrep wrote:

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

Is the reason why $3^k+3^k+3^k$ doesn't equal $3^2k$ is because exponential terms that are added or subtracted cannot be combined? I am reviewing the MGMT algebra book and didn't understand that concept at first.

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

30 Aug 2019, 16:52

Expert Reply

whollymoses wrote:

ScottTargetTestPrep wrote:

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

Is the reason why $3^k+3^k+3^k$ doesn't equal $3^2k$ is because exponential terms that are added or subtracted cannot be combined? I am reviewing the MGMT algebra book and didn't understand that concept at first.

Hi whollymoses,

The GMAT will sometimes test you on concepts that you know, but in ways that you are not used to thinking about. For example, I bet that you know how to do basic Arithmetic.

For example:
1 + 1 + 1 = 3(1)

In that same way, I bet that you can do basic Algebra too:
X + X + X = 3(X)

At this point, you probably recognize the pattern. Now, apply it to the math you asked about...
$3^k+3^k+3^k = 3(3^k$)

GMAT assassins aren't born, they're made,
Rich

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

31 Aug 2019, 20:15

Expert Reply

whollymoses wrote:

Is the reason why $3^k+3^k+3^k$ doesn't equal $3^2k$ is because exponential terms that are added or subtracted cannot be combined? I am reviewing the MGMT algebra book and didn't understand that concept at first.

Yes you are correct. The only thing you can do with 3^k + 3^k + 3^k is factor out 3^k, giving you:

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

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

01 Sep 2019, 07:09

Bunuel, VeritasKarishma why cant we use the exponents formula of (a^x)^c = c.a^x in this question. For eg : Can this (3^9)^3^9 be written as 3^9.3^9?
Any help on this would be appreciated. Thanks

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

01 Sep 2019, 09:31

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.

Given: $3^k + 3^k = (3^9)^{3^9}-3^k$

Asked: What is the value of k?

$3.3^k = 3^{9.3^9}$
$3^{k+1} = 3^{3^11}$
$k+1 = 3^11$
$k = 3^11 -1$

IMO 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]

02 Sep 2019, 04:47

Expert Reply

porwal1 wrote:

Bunuel, VeritasKarishma why cant we use the exponents formula of (a^x)^c = c.a^x in this question. For eg : Can this (3^9)^3^9 be written as 3^9.3^9?
Any help on this would be appreciated. Thanks

The exponent rule is:

$(a^b)^c = a^{bc}$
The exponents get multiplied.

It is not the same as $a^b * c$

We are given
$(3^9)^{3^9} = 3^{9*3^9}$

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

29 Feb 2024, 11:53

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

29 Feb 2024, 11:53

GMAT Problem Solving (PS) Questions

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