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

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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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11 Mar 2019, 06:33
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75% (hard)

Question Stats:

29% (01:11) correct 71% (01:26) wrong based on 38 sessions

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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

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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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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11 Mar 2019, 07:29
IMO E ,

after compare , we get √x= 2

means x = 4 (as x>0)
and also by trial , we see 1 satisfy the equation.

so sum = 5

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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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11 Mar 2019, 15:13
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

$$x > 0\,\,,\,\,\,\,{x^{\sqrt x }} = {x^2}\,\,\,\left( * \right)$$

$$?\,\,:\,\,{\rm{sum}}\,\,{\rm{of}}\,\,{\rm{roots}}$$

$$x = 1\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{inspection}}} \,\,\,\,\left\{ \matrix{ \,\,{x^{\sqrt x }} = {1^{\sqrt 1 }} = 1 \hfill \cr \,\,{x^2} = {1^2} = 1 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{first}}\,\,{\rm{root}}$$

$$0 < x < 1\,\,\,{\rm{or}}\,\,\,x > 1\,\,\,:\,\,\,\,\,$$

$$\left( * \right)\,\,\,\, \Rightarrow \,\,\,{x^{\sqrt x \, - \,2}} = 1 = {x^0}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{base}}\,\, \ne \,\,0,1, - 1} \,\,\,\,\,\sqrt x - 2 = 0\,\,\,\, \Rightarrow \,\,\,\,\,x = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{second}}\,\,{\rm{root}}$$

$$? = 1 + 4$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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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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Updated on: 12 Mar 2019, 17:38
Hello fskilnik!

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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11 Mar 2019, 23:02
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

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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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12 Mar 2019, 04:14
1
jfranciscocuencag wrote:
Hello fskilnik!

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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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?   [#permalink] 12 Mar 2019, 04:14
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