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

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Joined: 02 Sep 2009
Posts: 53020
If √√√(5x) = 6√(4x), , what is the range of the possible values of x?  [#permalink]

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28 Jan 2019, 01:43
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Difficulty:

55% (hard)

Question Stats:

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

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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

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

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

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