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

Question

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

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

Bunuel · Accepted Answer

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 .

Answer: B.

ezhilkumarank · Answer

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

Answer: 21 (B)

Yalephd · Answer

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

jakolik · Answer

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

ezhilkumarank · Answer

@ 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.

mainhoon · Answer

5^21 2^22 = 2 10^n
10^21 = 10^n
Hence n = 21

Posted from my mobile device

fluke · Answer

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"

blueseas · Answer

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

fameatop · Answer

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
Answer B

PareshGmat · Answer

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

KarishmaB · Answer

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

abhimahna · Answer

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

chetan2u · Answer

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

ScottTargetTestPrep · Answer

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

Answer: B

kanak2016 · Answer

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?

ScottTargetTestPrep · Answer

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.

raghav1708 · Answer

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

5^21 *   ^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

Abhishek009 · Answer

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)

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

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

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