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

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Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n? [#permalink]
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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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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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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5^21 2^22 = 2 10^n
10^21 = 10^n
Hence n = 21

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

Originally posted by blueseas on 04 Aug 2013, 11:31.
Last edited by blueseas on 04 Aug 2013, 11:36, 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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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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$$5^{21} * 4^{11} = 2*10^p$$

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

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

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

Equating powers, p = 21 = Answer B

Originally posted by PareshGmat on 26 Feb 2014, 21:58.
Last edited by PareshGmat on 15 Apr 2014, 01:56, 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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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]
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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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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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]
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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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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If 5^21 * 4^11 = 2 * 10^n, what is the value of n? [#permalink]
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

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

5^21 * [(2) ^2] ^11 = 2 * 10^n

5^21 * 2^22 = 2^1 * 10^n (Cancelling 2^1 from 2^22)

5^21 * 2^21 = 10^n (Same Exponents)

10^21 = 10^n

n=21

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

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

Or, $$5^{21} * 2^{22} = 2 * 10^n$$

Or, $$10^{21} * 2 = 2 * 10^n$$, Thus, $$n = 21$$, Answer must be (B)
Re: If 5^21 * 4^11 = 2 * 10^n, what is the value of n? [#permalink]
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