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Intern  Joined: 27 Feb 2012
Posts: 32
If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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Difficulty:   5% (low)

Question Stats: 85% (00:49) correct 15% (01:16) wrong based on 130 sessions

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If $$(\frac{2}{3})^n=(\frac{3}{2})^2$$, what is the value of n?

A. -2
B. -1
C. 0
D. 1
E. 2
Intern  Joined: 06 Mar 2012
Posts: 32
Re: If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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I was surprised to see this question in the 600-700 range. It's just testing the basic exponent property involving a negative sign.

(n)^-1 = (1/n)
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Joined: 02 Sep 2009
Posts: 58295
Re: If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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2
Edited the difficulty tag of this question.

jsphcal wrote:
If (2/3)^n=(3/2)^2, what is the value of n?

A. -2
B. -1
C. 0
D. 1
E. 2

Since $$(\frac{3}{2})^2=(\frac{2}{3})^{-2}$$ then we have that $$(\frac{2}{3})^n=(\frac{2}{3})^{-2}$$ --> $$n=-2$$.

For more on number theory and exponents check: math-number-theory-88376.html

DS questions on exponents: search.php?search_id=tag&tag_id=39
PS questions on exponents: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope it helps.
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Re: If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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A negative exponent in a fraction leads to the reciprocal of the fraction. Thus the negative exponent of 2 is -2, allowing (2/3)^(-2) = (3/2)^2.
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Re: If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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(2/3)^n = (3/2)^2
= (2/3)^-2

n = -2 (Answer = A)
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Re: If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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jsphcal wrote:
If (2/3)^n=(3/2)^2, what is the value of n?

A. -2
B. -1
C. 0
D. 1
E. 2

(2/3)^n = (3/2)^2

Observe that 3/2 = (2/3)^-1. Then:

(2/3)^n = ((2/3)^-1)^2

(2/3)^n=(2/3)^-2

n = -2

Alternate Solution:

Let’s distribute the exponents to the numerator and the denominator:

2^n/3^n = 3^2/2^2

If we cross multiply, we have:

2^(n + 2) = 3^(n + 2)

We see that the exponents of 2 and 3 are equal. This is only possible if n + 2 = 0. Then, n = -2.

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Re: If (2/3)^n = (3/2)^2, what is the value of n?  [#permalink]

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_________________ Re: If (2/3)^n = (3/2)^2, what is the value of n?   [#permalink] 13 Oct 2019, 01:44
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