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# If 5^21 * 4^11 = 2 * 10^n, what is the value of n?

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GMAT Date: 08-08-2011
If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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Updated on: 04 Oct 2017, 05:39
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Question Stats:

86% (01:02) correct 14% (01:39) wrong based on 550 sessions

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If $$5^{21} * 4^{11} = 2 * 10^n$$, what is the value of n?

A. 11
B. 21
C. 22
D. 23
E. 32

Originally posted by Yalephd on 08 Aug 2010, 20:12.
Last edited by Bunuel on 04 Oct 2017, 05:39, edited 4 times in total.
Edited the question
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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09 Aug 2010, 02:36
4
1
8
Yalephd wrote:
If 5^21 X 4^11 = 2x10^n. What is the value of n?

A. 11
B. 21
C. 22
D. 23
E. 32

When we have exponents on both sides it's good idea to get common base.

We have $$5^{21}*4^{11}=2*10^n$$. Little trick for this question: we can forget about 2's on both sides and concentrate on 5-s --> $$5^{21}*4^{11}=2*2^n*5^n$$ --> powers of 5 on both sides must be equal --> $$n=21$$.

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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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08 Aug 2010, 20:41
5
6
My attempt:

5^(21) * 4^(11) = 2*10^p

5^(21) * 2 ^ (22) = 2 * (2*5)^p

5^(21) * 2 ^ (22) = 2 * 2^(p) * 5^(p)

5^(21) * 2 ^ (22) = 2^(p+1) * 5^(p)

=> (p+1) = 22

=> p = 21

##### General Discussion
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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09 Aug 2010, 03:57
I got it now using ezhilkumarank's method. Thanks.
Bunuel, I don't follow your method. Are you saying that 5^21 * 4^11 = 2 * 2^21 * 5^21
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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09 Aug 2010, 05:15
1
Yalephd,

What Bunuel is trying to say is that save time by considering only 5

We have 10^p = (2*5)^p = 2^p * 5^p
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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14 Aug 2010, 15:12
1
vasili wrote:
Can somebody please put detailed explanation? Thanks.

If 5^21 x 4^11=2x10^n, what's n?

11
21
22
23
32

@ vasili
-- This question is already answered in one of the earlier post. Kindly search for the question once before posting.

5^21 * 4^11=2*10^n

= 5^21 * 4^11=2*(2*5)^n

= 5^21 * 4^11 = 2^(n+1) * 5^n

=> n = 21.
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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23 Aug 2010, 08:36
1
1
5^21 2^22 = 2 10^n
10^21 = 10^n
Hence n = 21

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Re: If 5^21*4^11 = 2*10^n, what is the value of n?  [#permalink]

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05 Feb 2011, 21:40
2
These types are usually solved by reducing the term to prime factors and few common rules of exponent.

5^21 * 4^11 = 2 * 10^n

Consider the bases first; 5,4,2,10
5 and 2 are already primes; 4 and 10 composites.
Reduce these to prime factors:
4=2*2
10=5*2
Place in the equation:

$$5^{21} * 4^{11} = 2 * 10^n$$
$$5^{21} * (2*2)^{11} = 2 * (2*5)^n$$
$$5^{21} * (2)^{11} *(2)^{11}= 2 * (2)^n*(5)^n$$
##$$(x*y)^m = x^m * y^m$$#### Rule of exponent##
$$5^{21} * (2)^{22} = (5)^n * 2^{(n+1)}$$
##$$x^m*x^n = x^{(m+n)}$$## Rule of exponent##

Compare LHS and RHS
n+1=22
n=21

Ans: "B"
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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Updated on: 04 Aug 2013, 10:36
2
njkhokh wrote:
If 5^21 x 4^11 = 2 x 10^n what is the value of n?

A) 11
B) 21
C) 22
D) 23
E) 32

$$5^21$$$$*4^11$$ $$= 2*10^n$$

$$5^21* 2^22 = 2 *10^n$$ ( AS $$4^11= (2^2)^11$$)

$$(5.2)^21 * 2 = 2 * 10^n$$

HENCE n = 21
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Originally posted by blueseas on 04 Aug 2013, 10:31.
Last edited by blueseas on 04 Aug 2013, 10:36, edited 1 time in total.
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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04 Aug 2013, 10:36
4
njkhokh wrote:
If 5^21 x 4^11 = 2 x 10^n what is the value of n?
A) 11
B) 21
C) 22
D) 23
E) 32

5^21 x 4^11 = 2 x 10^n
5^21 x 2^22 = 2 x 10^n
2 x 5^21 x 2^21 = 2 x 10^n
2 x 10^21 = 2 x 10^n

Thus n = 21
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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Updated on: 15 Apr 2014, 00:56
4
$$5^{21} * 4^{11} = 2*10^p$$

$$5^{21} * 2^{22} = 2*10^p$$

$$2 * 2^{21} * 5^{21} = 2*10^p$$

$$2 * 10^{21} = 2*10^p$$

Equating powers, p = 21 = Answer B

Originally posted by PareshGmat on 26 Feb 2014, 20:58.
Last edited by PareshGmat on 15 Apr 2014, 00:56, edited 1 time in total.
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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14 Apr 2014, 17:23
2
1
hi, all!

quick exponent question which seems easy but i cant seem to get my head around it - reallly appreciate your help!!

If 5^21 x 4^11 = 2 x 10^n,, what is value of n?

answer is 21, but honestly cant remember quite how to get here - thanks again for any help!

Thumb rule for any exponents question: Deal only in prime numbers.

$$5^{21} * 4^{11} = 2 * 10^n$$

$$5^{21} * 2^{22} = 2 * 2^n * 5^n$$

$$5^{21} * 2^{22} = 2^{n+1} * 5^n$$

Now match the exponents of same prime numbers. Power of 5 on both sides should be equal.
21 = n
or
Power of 2 on both sides should be equal.
22 = n+1
n = 21
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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27 Aug 2016, 22:53
SharmaShruti wrote:
5^21 X 4^11 = 2 X 10^n. What is the value of n?
a) 11
b) 21
c) 22
d) 23
e)32

The given expression $$5^21 * 4^(11) = 2 * 10^n$$ can be written as $$5^21 * 2^22 = 2 * 5^n * 2^n$$

or $$5^21 * 2^22 = 5^n * 2^(n+1)$$

Comparing the powers of 2 on both sides, we will have $$2^22 = * 2^(n+1)$$

or 22=n+1 or n=21. hence B
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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28 Aug 2016, 05:38
1
SharmaShruti wrote:
5^21 X 4^11 = 2 X 10^n. What is the value of n?
a) 11
b) 21
c) 22
d) 23
e)32

Hi,
One does not require to simplify anything..
On left hand side, the power of 5 is 21.....
But on right side it is part of 10.. so power of 10 has to be same as that of 5..
Ans 21
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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06 Oct 2017, 09:53
1
Yalephd wrote:
If $$5^{21} * 4^{11} = 2 * 10^n$$, what is the value of n?

A. 11
B. 21
C. 22
D. 23
E. 32

We can simplify the given equation:

5^21 x 2^22 = 2^1 x 10^n

5^21 x 2^21 = 10^n

10^21 = 10^n

21 = n

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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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14 Feb 2020, 09:32
ScottTargetTestPrep wrote:
Yalephd wrote:
If $$5^{21} * 4^{11} = 2 * 10^n$$, what is the value of n?

A. 11
B. 21
C. 22
D. 23
E. 32

We can simplify the given equation:

5^21 x 2^22 = 2^1 x 10^n

5^21 x 2^21 = 10^n

10^21 = 10^n

21 = n

Hi Scott,

i like the way you solved the above problem. Just a small question.
In the above simplification, did you divide 2^22 by 2^1(on the left hand side) to make it 2^21?
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?  [#permalink]

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16 Feb 2020, 17:44
2
kanak2016 wrote:
ScottTargetTestPrep wrote:
Yalephd wrote:
If $$5^{21} * 4^{11} = 2 * 10^n$$, what is the value of n?

A. 11
B. 21
C. 22
D. 23
E. 32

We can simplify the given equation:

5^21 x 2^22 = 2^1 x 10^n

5^21 x 2^21 = 10^n

10^21 = 10^n

21 = n

Hi Scott,

i like the way you solved the above problem. Just a small question.
In the above simplification, did you divide 2^22 by 2^1(on the left hand side) to make it 2^21?

That's exactly what I did. I divided each side by 2^1 so that we are left with 10^n on one side and subtracted the exponents of 2 on the other side to obtain 2^{22 - 1} = 2^21.
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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n?   [#permalink] 16 Feb 2020, 17:44