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If x^a*x^b = x^c, is what is c/(a+b)?

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If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   Updated on: 10 May 2020, 10:59
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If \(x^a*x^b = x^c,\) is what is \(\frac{c}{(a+b)}\)?

(1) a, b are integers
(2) a, and b are positive numbers
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E

Originally posted by GMATBusters on 09 May 2020, 18:53.
Last edited by GMATBusters on 10 May 2020, 10:59, edited 2 times in total.
Renamed the topic and edited the question.
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   09 May 2020, 18:55
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Official Solution:




if a+b = c, then \(x^a*x^b = x^c\)
but vice versa is tricky.
  • if x = 1 or 0, then we can have \(x^a*x^b = x^d\), where d is any number
  • similarly, if x = -1, x^n will always = 1 , if n is even
  • hence if, \(x^a*x^b = x^c\). it is not necessary that c = a+b


If \(x^a*x^b = x^c,\) is what is \(\frac{c}{(a+b)}\)?

(1) a, b are integers
if x \(\neq{0}\) 1, 0, -1. then we can say c=a+b, but
if x = 1, 0, -1. then we can't say c=a+b
Not Sufficient

(2) a, and b are positive numbers
if x \(\neq{0}\) 1, 0, -1. then we can say c=a+b, but
if x = 1, 0, -1. then we can't say c=a+b
Not Sufficient

Even after combining both statements 1 & 2, we can find value of \(\frac{c}{(a+b)}\)
Hence, Answer is E
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   Updated on: 09 May 2020, 21:36
If x^a∗x^b=x^c, is what is c/(a+b)?
We need unique value of c/a+b.
x^a∗x^b=x^c
x^a+b=x^c
a+b =c
1) a, b are integers
If a = 2 and b = 3, then c = 5. So, 5/5 = 1
If a = 5, b = 7, then c = 12, So, 12/12=1
If a = -1, b = 2, then c = 1, 1/1= 1.
Sufficient.

2) a, and b are positive numbers
If a = 2 and b = 3, then c = 5. So, 5/5 = 1
If a = 1/2, b = 2, then c=2.5 (5/2), so 5/2 divided by 5/2 = 1
If a = 3/2, b = 2/3, then c=13/6, so 13/6 divided by 13/6 = 1
So sufficient.

Ans. D

Originally posted by Raxit85 on 09 May 2020, 19:03.
Last edited by Raxit85 on 09 May 2020, 21:36, edited 1 time in total.
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   09 May 2020, 19:28
As per exponents rule, \(x^a*x^b= x^a+x^b\).
So \(x^a+x^b=x^c\)
Here there are two situations possible. If a and b have different signs adding upto zero, there can be a possibility like-
\(1^-2* 1^2= 1^1\)
\(1^0= 1^1\)
However c/a+b= 1/0 here is undefined because the denominator is zero.

The other case can be where a+b=c
And here c/a+b= 1
Thus, we need to know the positive/ negative nature of a and b.

1) Does not give us the required information regarding the positive/ negative nature of a and b. No sufficient.
2) a and b are positive integers. So the answer would be 1. Sufficient.

Answer: B
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   Updated on: 10 May 2020, 09:42
given
x^a* x^b = x^c
a+b=c
need to find c/a+b
#1
a,b are integers
we can have
(-1)^1*(-1)^2 = (-1)^√1
(-1)^3 = (-1)^√1
we get c/(a+b) =√1 /3 or c can be any odd value
or
2^2*2^3 = 2^5
2^5=2^5
we get c/(a+b) = 5/5 ; 1
insufficient

#2 a, and b are positive numbers

again same as #1
(-1)^1*(-1)^2 = (-1)^√1
(-1)^3 = (-1)^√1
we get c/(a+b) =√1 /3
or
2^2*2^3 = 2^5
2^5=2^5
we get c/(a+b) = 5/5 ; 1
insufficient
From 1&2 nothing can be determined
OPTION E

If x^a∗x^b=x^c,, is what is c/(a+b)?
1) a, b are integers
2) a, and b are positive numbers

Originally posted by Archit3110 on 09 May 2020, 19:30.
Last edited by Archit3110 on 10 May 2020, 09:42, edited 1 time in total.
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   09 May 2020, 20:11
Answer E
c/a+b=?
State1:Not sufficient
x^a+b-c=1
case 1, x^2+2-4=1
c/a+b=1
case 2, 1^2+3-1=1
c/a+b=1/5
State:2not sufficient
a,b>0
x^a+b-c=1
case 1, x^2+2-4=1
c/a+b=1
case2, let x=1, 1^2+3-1=1
c/a+b=1/5
statement 1+2,
both case apply in this case also
Hence not sufficient
Answer E
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If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   Updated on: 10 May 2020, 18:51
If \(x^a∗x^b=x^c,\) is what is c/(a+b)
If x=0 or 1; a,b & c can take multiple values.

1) a, b are integers
Since x may be 0 or 1 or -1
NOT SUFFICIENT

2) a, and b are positive numbers
Since x may be 0 or 1 or -1
NOT SUFFICIENT

(1) + (2)
1) a, b are integers
2) a, and b are positive numbers
Since x may be 0 or 1 or -1
NOT SUFFICIENT

IMO E


Originally posted by Kinshook on 09 May 2020, 20:19.
Last edited by Kinshook on 10 May 2020, 18:51, edited 2 times in total.
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   09 May 2020, 21:46
Answer is D.

Statement 1 is sufficient to answer the question.
Statement 2 is sufficient to answer the question.
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   10 May 2020, 02:19
x^a*x^b = x^c
--> x^(a+b) = x^c

Q. c/(a+b)?

(1) a, b are integers
If a=1, b=-1, c=0, and x is not 1, then c/(a+b) is undefined.
If a=1, b=2, c=3, and x is not 1, then c/(a+b)=1
NOT SUFFICIENT

(2) a, and b are positive numbers
If a=1, b=2, c=3, and x is not 1, then c/(a+b)=1
If a=1, b=2, c=9, and x=1, then c/(a+b)=1/3
We need to know x
NOT SUFFICIENT

Combined
If a=1, b=2, c=3, and x is not 1, then c/(a+b)=1
If a=1, b=2, c=9, and x=1, then c/(a+b)=1/3
We need to know x to determine exact value of c/(a+b)
NOT SUFFICIENT

FINAL ANSWER IS (E)

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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   10 May 2020, 03:34
If x^a∗x^b=x^c,, is what is c/(a+b)?

(1) a, b are integers
(2) a, and b are positive numbers

Simplifying the qs stem => \(x^{a+b}\) = \(x^c\).

If x = 0, it doesn't matter what a, b and c are. Since LHS will always be equal to RHS.
Or
If x=2 => a+b = c in which case the value of c/(a+b) would be 1 if c or (a+b) is not equal to 0

And we don't have information about nature of x in either statements.

Answer - E
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink] If x^a*x^b = x^c, is what is c/(a+b)?   10 May 2020, 05:15
If xa∗xb=xc,xa∗xb=xc, is what is c/(a+b)c/(a+b)?
1) a, b are integers
2) a, and b are positive numbers

Since c=a+b, \(c/(a+b) \) , \(c/(a+b)\) = 1
However, c must not equal to 0 as it will be undefined (0/0 is undefined)

1) if a and b are integers, there will be a probability that a + b = 0
For example, a may be -2 and b might be 2. \(\frac{c}{{a+b}}=\frac{0}{{0}}\) = not defined
Hence, A and D are incorrect

2) If both a and b are integers, c will always be 1. hence, B is correct
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Re: If x^a*x^b = x^c, is what is c/(a+b)? [#permalink]
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