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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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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

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

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

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

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

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

$$? = 1 + 4$$

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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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]
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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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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