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If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2? [#permalink]
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Hello fskilnik!

Why did you start with "inspection" 1?

What I did was:

\(\,{x^{\sqrt x }} = {x^2}\)

\(({\sqrt x } = 2)^2\)

\(x = 4\)

Am I totally wrong?

Regards!

Thank you fskilnik !

Originally posted by jfranciscocuencag on 11 Mar 2019, 21:00.
Last edited by jfranciscocuencag on 12 Mar 2019, 17:38, edited 1 time in total.
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Re: If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2? [#permalink]
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fskilnik wrote:
GMATH practice exercise (Quant Class 12)

If \(x>0\), what is the sum of the roots of the equation \(\,{x^{\sqrt x }} = {x^2}\) ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


When the base is 0 or 1 or -1, it is possible that the two terms are equal.
But x cannot be 0 here because 0^0 is not defined.
If x = 1. the two terms are equal.
x cannot be -1 here since x > 0

Another way the two terms will be equal is if the exponents are equal.
\(\sqrt{x} = 2\)
\(x = 4\)

Sum of the roots = 1 + 4 = 5

Answer (E)
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If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2? [#permalink]
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jfranciscocuencag wrote:
Hello fskilnik!

Why did you start with "inspection" 1?

What I did was:

\(\,{x^{\sqrt x }} = {x^2}\)

\(({\sqrt x } = 2)^2\)

\(x = 4\)

Am I totally wrong?

Regards!

Hi jfranciscocuencag !

Let me add other (I believe) nice details to complement Karishma´s important contribution.

(I am glad you have joined us, VeritasKarishma! I hope everything is well with you and your family!)

Given a fixed constant (say) \(b\), we have learned at school that \(b^x = b^y\) implies \(x=y\), although this is not, in general, true.

Examples: (i) \(0^1 = 0^2\) but \(1=2\) is false. (ii) \((-1)^0 = (-1)^2\) but \(0=2\) is false...

The true statement is the following: \(b^x = b^y\) implies \(x=y\) when the base \(b\) is different from -1, 0, and 1 (as Karishma properly mentioned).

This is a consequence of the injectivity of the exponential function when the base \(b\) is positive and different from 1, meaning: different powers will give different values when the base is the same but different from -1,0, and 1.

That´s exactly the reason I had to check the case of the base equals to 1 separately, and I did by inspection, of course.

Please note that Karishma and I were able to "impose equal powers" when we were in the blue case above...

I hope things are clear now!

Regards and success in your studies,
Fabio.
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Re: If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2? [#permalink]
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Re: If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2? [#permalink]
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