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655-705 (Hard)|   Exponents|               
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avigutman
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If x^2 = 2^x, what is the value of x ? 03 Jul 2022, 14:31
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IanStewart
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I donot get it. If the irrational value works in both question stem and statement 2 why does it not work when those two are used together IanStewart??

The irrational value that works in the stem is not the same as the irrational value that works in Statement 2.
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Just to add an analogy here, Elite097: imagine if the free info in the stem told us that Avi is either in Canada or in Europe, and statement (2) told us that Avi is either in Canada or in Asia. Using all of that information, you can conclude that I'm in Canada.
In other words, as we gain more information, we can (and should) use the new information to eliminate some of the scenarios that were previously possible.
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If x^2 = 2^x, what is the value of x ? 05 Jul 2022, 04:24
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If x^2 = 2^x, what is the value of x ? 27 Dec 2022, 17:13
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chetan2u
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If \(x^2 = 2^x\), what is the value of x ?


(1) \(2x = (\frac{x}{2})^3\)

(2) \(x = 2^x-2\)

Hi..

If \(x^2 = 2^x\), what is the value of x ?


(1) \(2x = (\frac{x}{2})^3\)
\(x*(\frac{x^2}{8}-2)=0\)
we get x as 0, 4 or -4..
when we substitute value in \(x^2 = 2^x\), ONLY x = 4 is possible..
sufficient


(2) \(x = 2^{x-2}\)
or \(x=2^{x-2}\)...
2^x has to be multiple of 2 and so also x should be in power of 2 that is 2,4,8,16...AND x will fit in only at x=4 as \(2^{4-2}\)
ONLY when x is 4 both sides are 4..
suff

D

carcass i believe statement II should be \(x=2^{x-2}\).. here possible value is 4 SAME as statement I.. pl relook into Q
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chetan2u

Thank you for your helpful explanation. I am just a bit confused on expressing x=2^(x−2) in different ways...
To confirm, does 2^(x-2) = (2^x)*(2^-2) ?

Thank you!
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If x^2 = 2^x, what is the value of x ? 27 Dec 2022, 23:06
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chetan2u
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If \(x^2 = 2^x\), what is the value of x ?


(1) \(2x = (\frac{x}{2})^3\)

(2) \(x = 2^x-2\)

Hi..

If \(x^2 = 2^x\), what is the value of x ?


(1) \(2x = (\frac{x}{2})^3\)
\(x*(\frac{x^2}{8}-2)=0\)
we get x as 0, 4 or -4..
when we substitute value in \(x^2 = 2^x\), ONLY x = 4 is possible..
sufficient


(2) \(x = 2^{x-2}\)
or \(x=2^{x-2}\)...
2^x has to be multiple of 2 and so also x should be in power of 2 that is 2,4,8,16...AND x will fit in only at x=4 as \(2^{4-2}\)
ONLY when x is 4 both sides are 4..
suff

D

carcass i believe statement II should be \(x=2^{x-2}\).. here possible value is 4 SAME as statement I.. pl relook into Q

chetan2u

Thank you for your helpful explanation. I am just a bit confused on expressing x=2^(x−2) in different ways...
To confirm, does 2^(x-2) = (2^x)*(2^-2) ?

Thank you!
Show more

Hi

\(2^{x-2}=2^x*2^{(-2)}=2^x*\frac{1}{2^2}=\frac{2^x}{2^2}\)
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If x^2 = 2^x, what is the value of x ? 17 Jun 2023, 06:47
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avigutman gmatophobia vv65

Would you mind checking if my approach is sound? I'm unsure mainly because it does not seem to produce 0 or negative values for x.

Free info: x^2 = 2^x

S1. 2x = x^3/2^3

Isolating x...
x^2 = (2^3)(2)
x^2 = 2^4

Powering both sides by 1/2 to find x...
x = 2^2 = 4

SUFFICIENT

S2. x = 2^(x-2)
x = 2^x/2^2

Replacing 2^x with x^2...
x = x^2/2^2

Isolating x...
x = 2^2 = 4

SUFFICIENT
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If x^2 = 2^x, what is the value of x ? 17 Jun 2023, 16:51
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achloes
avigutman gmatophobia vv65

Would you mind checking if my approach is sound? I'm unsure mainly because it does not seem to produce 0 or negative values for x.

Free info: x^2 = 2^x

S1. 2x = x^3/2^3

Isolating x...
x^2 = (2^3)(2)
x^2 = 2^4

Powering both sides by 1/2 to find x...
x = 2^2 = 4

SUFFICIENT

S2. x = 2^(x-2)
x = 2^x/2^2

Replacing 2^x with x^2...
x = x^2/2^2

Isolating x...
x = 2^2 = 4

SUFFICIENT
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Hi achloes, I highlighted your errors above.
In general, I advise students to use arithmetic language when they manipulate equations and inequalities. So, use "dividing the equation by a factor of x" rather than "isolating x". That way you're more likely to realize that you can't perform that operation without acknowledging the possibility that x=0.
Your other error was not accounting for the possibility that x is negative when you took the square root of the equation.
If we know that x^2=9, for example, we can only infer that x is 3 units away from zero, but there's no way for us to know which side. In other words:
x^2 = 9 ---> |x| = 3
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If x^2 = 2^x, what is the value of x ? 18 Jun 2023, 02:11
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avigutman
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avigutman gmatophobia vv65

Would you mind checking if my approach is sound? I'm unsure mainly because it does not seem to produce 0 or negative values for x.

Free info: x^2 = 2^x

S1. 2x = x^3/2^3

Isolating x...
x^2 = (2^3)(2)
x^2 = 2^4

Powering both sides by 1/2 to find x...
x = 2^2 = 4

SUFFICIENT

S2. x = 2^(x-2)
x = 2^x/2^2

Replacing 2^x with x^2...
x = x^2/2^2

Isolating x...
x = 2^2 = 4

SUFFICIENT
Hi achloes, I highlighted your errors above.
In general, I advise students to use arithmetic language when they manipulate equations and inequalities. So, use "dividing the equation by a factor of x" rather than "isolating x". That way you're more likely to realize that you can't perform that operation without acknowledging the possibility that x=0.
Your other error was not accounting for the possibility that x is negative when you took the square root of the equation.
If we know that x^2=9, for example, we can only infer that x is 3 units away from zero, but there's no way for us to know which side. In other words:
x^2 = 9 ---> |x| = 3
Show more

Flawless explanation - thank you!
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If x^2 = 2^x, what is the value of x ? 18 Jun 2023, 18:09
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achloes
Would you mind checking if my approach is sound? I'm unsure mainly because it does not seem to produce 0 or negative values for x.
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Your answer for this question was right; that was luck. AviGutman has already explained the possibilities you neglected.

We must always remember these points:
x^2 = 4 does not always mean x=2. It could mean x = -2.
2x = xy does not always mean y=2. It could mean x=0.
Neglecting these possibilities will often (usually?) lead to a wrong answer.
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If x^2 = 2^x, what is the value of x ? 02 Sep 2025, 07:03
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