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If x and y are positive integers, what is the value of x^y?

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Bunuel
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If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 01:18
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If x and y are positive integers, what is the value of \(x^y\)?


(1) \(y^x= 9\)

(2) \(x^{(2y)}= 64\)


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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 01:40
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1) Y^x = 9
Case 1: 9^1 = 9
Case 2: 3^2 = 9
(Insufficient)

2) X^2y = 64
Prime factors of 64 = 2^6
X^2y = 2^6
On comparing both sides.
X = 2
Y = 3
Therefore x^y = 8
(Sufficient)

IMO B

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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 02:19
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Premise - X,Y = POSITIVE INTEGERS
X^Y=?

STAT1 - Y^X =9
CASES -
- X=2, Y=3 OR X=1,Y=9
Both give different values for X^Y ,
hence not sufficient

STAT 2
X^2y=2^6
now possible cases with both x and y integers are
x=2 Y= 3 : and X=8 , Y =1
Both give same value for X^Y=8
hence sufficient
ANSWER B

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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 02:25
Given x and y are positive integers.
stmt 1: y^x=9
possibilities-
y=3, x=2
y=9, x=1
stmt 1 - insufficient because we do not have a definite answer.
stmt 2: x^(2y)=64
x^(2y)=64 --> x^y=8 (because x and y are +ve)
possibilities-
x=2, y=3
x=8, y=1
stmt 2- insufficient because we do not have a definite answer.

Combining both statements,
we see an overlap of values,
x=2, y=3.
Therefore answer, C
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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 07:50
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Given: x and y are positive integers => x,y -> [1,2,3......infinity)

Statement 1:
y^x = 9
(i) 9^1 = 9
=> y = 9, x = 1
=> x^y = 1^9 = 1

(ii) 3^2 = 9
=> y = 3, x = 2
=> x^y = 2^3 = 8

=> Statement 1 is insufficient

Statement 2:
x^2y = 64
(i) x^2y = 8^2
=> x = 8, y = 1
=> x^y = 8

(ii) x^2y = 2^6
=> x = 2, y = 3
=> x^y = 8

No other possibility is there.

=> Statement 2 is sufficient


Answer: B
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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  Updated on: 31 Jul 2020, 06:07
Quote:
If x and y are positive integers, what is the value of x^y?

(1) y^x=9

Step 1: Understanding statement 1 alone
\(y^x\)=9 = \(3^2\)
Hence, possible values are
When\( y^x\)=9; x=1, y = 8
When \(y^x= 3^2\); x = 2, y = 3
Insufficient

Quote:
(2)\( x^{2y}\)=64

Step 2: Understanding statement 2 alone
\(x^{2y}\)=64 = \(2^6 = 8^2\)
When \(x^{2y}\)= \(2^6\),=; x= 2, y = 3,\( x^y\) = 8
When\( x^{2y} = 8^2\); x = 8, y = 1, \(x^y\) = 8
Sufficient

IMO B
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Originally posted by QuantMadeEasy on 30 Jul 2020, 10:17.
Last edited by QuantMadeEasy on 31 Jul 2020, 06:07, edited 1 time in total.
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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 12:50
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Expert Reply

Solution


Step 1: Analyse Question Stem


    • x and y are positive integers.
    • We need to find the value of \(x^y\)

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE


Statement 1: \(y^x = 9\)
    • There can be following cases:
      o Case 1: \(y^x = 9^1\)
         In this case, x = 1 and y = 9
          • Therefore, \(x^y =1^9 = 1\)
      o Case 2: \(y^x = 3^2\)
         In this case, x = 2 and y = 3
          • Therefore, \(x^y =2^3 = 8\)
    • We are getting two different results.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.

Statement 2: \(x^{(2y)} =64\)
• We have, \(x^{(2y)} =64⟹ (x^y)^2 = 64 ⟹ x^y = 8\) [since x and y both are positive \(x^y\) cannot be negative or -8]
Hence statement 2 is sufficient.
Thus, the correct answer is Option B.
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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 15:41
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1) When y = 9 and x = 1, x^y = 1. Again when y = 3, x = 2, x^y = 2^3 = 8. Two different answers. Noot sufficient.

2) x^2y = 64. 64 can be expressed as 64^1 (which is not possible here as y cannot be 1/2). 8^2 (possible when x = 8, y =1, x^y = 8), or 4^3 (not possible as y cannot be 3/2). Can ignore (-8)^2 as x and y have to be positive integers. So one unique value found. Sufficient.

B is the answer.
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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 18:28
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Quote:
If x and y are positive integers, what is the value of xyxy?


(1) yx=9yx=9

(2) x(2y)=64x(2y)=64


(1) insufic
y^x=9: {3^2,9^1}
x^y: {2^3=8}{1^9=1}

(2) sufic
x^2y=64={64^1,8^2,4^3,2^6}
2y={1,2,3,6}
y={1/2,1,3/2,3}={1,3}
x={8,2}
x^y: {8^1=8}{2^3=8}

ans (B)
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Re: If x and y are positive integers, what is the value of x^y? [#permalink] New post  30 Jul 2020, 20:06
C is the answer
(1) not sufficient. There are two scenarios; x=1 and y=9 or x=2 and y=3
(2) not sufficient. There are two scenarios; x = 2 and y =3 or x=8 and y=1

combine (1) and (2), we have x=2 and y=3
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