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If √√√(5x) = 6√(4x), , what is the range of the possible values of x?

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If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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New post 28 Jan 2019, 01:43
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

60% (02:56) correct 40% (02:07) wrong based on 37 sessions

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Re: If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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New post 28 Jan 2019, 01:55
\((((5x)^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}} = (2x)^{\frac{1}{6}}\)
\((5x)^{\frac{1}{8}} = (2x)^{\frac{1}{6}}\)
Raising to the power of 24 give
\((5x)^{3} = (2x)^{4}\)
\((5)^{3}(x)^{3} = (2)^{4}(x)^{4}\)
Either x=0 or dividing by \(x^3\)
\((5)^{3} = (2)^{4}(x)\)
\(x = \frac{125}{16}\)

hence range \(= \frac{125}{16} - 0 = \frac{125}{16}\)

IMO None of the options
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Re: If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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New post 28 Jan 2019, 02:01
4d wrote:
\((((5x)^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}} = (2x)^{\frac{1}{6}}\)
\((5x)^{\frac{1}{8}} = (2x)^{\frac{1}{6}}\)
Raising to the power of 24 give
\((5x)^{3} = (2x)^{4}\)
\((5)^{3}(x)^{3} = (2)^{4}(x)^{4}\)
Either x=0 or dividing by \(x^3\)
\((5)^{3} = (2)^{4}(x)\)
\(x = \frac{125}{16}\)

hence range \(= \frac{125}{16} - 0 = \frac{125}{16}\)

IMO None of the options


It's \(\sqrt{\sqrt{\sqrt{5x}}} = \sqrt[6]{4x}\). Edited.
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Re: If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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New post 28 Jan 2019, 02:09
Bunuel wrote:
If \(\sqrt{\sqrt{\sqrt{5x}}} = \sqrt[6]{4x}\), what is the range of the possible values of x?
A. 0
B. 1/4
C. 125/256
D. 1/2
E. 131/256


5x ^ 1/8 = 4x^1/6

Multiplying both powers by 24

{5x}^3 = {\(2^2\) * x} ^4

{2^2 * x} ^ 4 - {5x}^3 = 0

x^3 ( 256 x - 125) = 0

Range = HT - LT

125/256 - 0

C
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Re: If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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New post 28 Jan 2019, 02:48
1
Bunuel wrote:
If \(\sqrt{\sqrt{\sqrt{5x}}} = \sqrt[6]{4x}\), what is the range of the possible values of x?


A. 0
B. 1/4
C. 125/256
D. 1/2
E. 131/256



(5x)^1/8 = (4x)^1/6

raising to 24 on both sides
(5x)^3= (4x)^4
125* x^3 = 256* x^4
x= 125/256
IMO C
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Re: If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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New post 28 Jan 2019, 03:00
\(((4x)^\frac{1}{6})^{2*2*2}\) = 5x

\(((4x)^\frac{4}{3})\) = 5x

\(((x)^\frac{4}{3})\) = \(\frac{{5x}}{4^{\frac{4}{3}}}\)

\(x^{\frac{1}{3}} = \frac{5}{4^{\frac{4}{3}}}\)

Cubing on both sides,

\(x = (\frac{5^3}{4^4})\)

x = 125/256

OPTION: C
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Re: If √√√(5x) = 6√(4x), , what is the range of the possible values of x?   [#permalink] 28 Jan 2019, 03:00
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