What is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?

Question

What is the value of  0.1 + 0.1^{\frac{1}{m}} + 0.1^{\frac{1}{n}} ?

(1)  \frac{1}{m}+ \frac{1}{n} = \frac{4}{3}

(2) mn = 3.

chetan2u · Answer

0.1 + 0.1^{\frac{1}{m}} + 0.1^{\frac{1}{n}}=\frac{1}{10} + \frac{1}{10}^{\frac{1}{m}} + \frac{1}{10}^{\frac{1}{n}}=\frac{10^{m+n}+10^{m+1}+10^{n+1}}{10^{m+n+1}} 
So, we require values of m and n..

When we combine statement 1 and statement 2, we get m and n as 1 and 3 in any order. When we substitute (m =3 and n=1) or (m=1 and n=3), we will get the same answer.
 \frac{10^{m+n}+10^{m+1}+10^{n+1}}{10^{m+n+1}}=\frac{10^{1+3}+10^{1+1}+10^{3+1}}{10^{1+3+1}}=\frac{10^4+10^2+10^4}{10^5}=\frac{20100}{100000}=\frac{201}{1000} =0.201

C

Poorvi05 · Answer

Can someone please explain this in detail!?

Posted from my mobile device

ocelot22 · Answer

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What is the value of .1 +.1^1/m +.1^1/n

We need to know the values of m and n to evaluate the above expression

(1) 1/m +1/n = 4/3. We can several combinations of values for m and n that will give us different values for our expression

let 1/m = 1/3 and let 1/n =1, then our expression becomes .1+.1^1/3 + .1 = .2 +.1^1/3

Let 1/m =2/3 and let 1/n =2/3, then our expression becomes .1 +.1^2/3 +.1^2/3

Since we get two different values for our expression we can mark NS

(2) mn = 3

Let m =1 and n =3. Then our expression becomes .1 + .1^1 + .1^3 = .1+.1 + .001 = .201

Let m = 3/2 and n =2. Then our expression becomes .1 + .1^3/2 + .1^2 = .11 + .1^3/2

Since we get two different values for our expression we can mark NS

(1) and (2). This gives us two equations that we can use to solve for m and n. From (2) m =3/n. We can substitute into (1) to get 1/(3/n) + 1/n =4/3.

This is 1 variable in 1 equation, so we can solve for n, which allows us to solve for m. Sufficient.

FTGreco · Answer

Since we have m and n as exponents in two separated terms from the equation, and those terms are being added together (so we cannot "merge" them like if they were being multiplied), we will need to figure out a way to find both  m  as well as  n .

(1) gives us one equation with 2 variables. This is not enough to solve for both of them, and thus is insufficient.

(2) gives us, again, one equation with 2 variables. Insufficient.

(1) + (2) gives us two equations for two variables, which can be solved.  \frac{1}{m}+ \frac{1}{n} = \frac{m + n}{mn} = \frac{4}{3} .
From this,  m+n = 4 , and  m = (4 - n) .

m * n = 3

(4 - n) * n = 3

n^2 - 4n + 3 = 0

n = 3  or  n = 1 .

If n = 3, m = 1. On the other hand, if n = 1, m = 3.

Now, since the terms to the power of  \frac{1}{m}  and  \frac{1}{n}  are both at the same base ( 0.1 ), it doesn't matter whether  (m,n) = (1,3)  or  (m,n) = (3,1) . Both will give the same final result, and this means that (1) and (2) together are sufficient.

Answer is C.

anmolgmat14 · Answer

If we Consider statement 1,
adding both the terms gives us the equation M+N/MN=4/3

Doesn't it tells us that M+N=4
& 
MN=3

SaquibHGMATWhiz · Answer

Solution:      
  Pre Analysis:  
  We are asked the value of  0.1 + 0.1^{\frac{1}{m}} + 0.1^{\frac{1}{n}}  
 This cannot be simplified in any way where any one statement will be sufficient 
  We need the value of both m and n  to get the answer

Statement 1:    \frac{1}{m}+ \frac{1}{n} = \frac{4}{3} 
 ⇒\frac{n+m}{mn}=\frac{4}{3} 
  We cannot get the value of m and n from this 
 Thus,  statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2:   mn = 3
  We cannot get the value of m and n from this 
 Thus,  statement 2 alone is also not sufficient and we can eliminate option B

Combining:   
  From statement 1, we have  \frac{n+m}{mn}=\frac{4}{3} 
 ⇒\frac{n+m}{3}=\frac{4}{3}  (from statement 2)
 ⇒n+m=4 
 ⇒\frac{3}{m}+m=4 
 ⇒\frac{3+m^2}{m}=4 
 ⇒m^2-4m+3=0  
 The  values of m are 3 and 1  
 When m = 3,  n=\frac{3}{m}=\frac{3}{3}=1  
 When m = 1,  n=\frac{3}{m}=\frac{3}{1}=3

When (m, n) = (3, 1), the value of  0.1 + 0.1^{\frac{1}{3}} + 0.1^{\frac{1}{1}}=0.2+0.1^{\frac{1}{3}}   When (m, n) = (1, 3), the value of  0.1 + 0.1^{\frac{1}{1}} + 0.1^{\frac{1}{3}}=0.2+0.1^{\frac{1}{3}}

Hence the right answer is    Option C

What is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?

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What is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?

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