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# If x^9 = 9^9^9, what is the value of x?

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Math Expert
Joined: 02 Sep 2009
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If x^9 = 9^9^9, what is the value of x?  [#permalink]

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30 Jul 2019, 07:50
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28% (01:03) correct 72% (00:48) wrong based on 134 sessions

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If $$x^9 = 9^{9^9}$$, what is the value of $$x$$?

A. $$9^8$$

B. $$9^9$$

C. $$9^{8^9}$$

D. $$9^{9^8}$$

E. $$9^{9^9}$$

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Joined: 15 Jul 2016
Posts: 105
Location: India
GMAT 1: 690 Q48 V36
If x^9 = 9^9^9, what is the value of x?  [#permalink]

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30 Jul 2019, 20:50
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1
x^9 = 9^(9^9)

To simplify understanding :

x^9 = 9^(9*9*9*9*9*9*9*9*9)

Taking L.H.S's power of 9 to the right side of the equation For example : [x^2 = 4 => square Root of 4 or 4^(1/2)]

x = 9^(9*9*9*9*9*9*9*9*9/9)

=> x = 9^(9*9*9*9*9*9*9*9) => 9^(9^8)

Ans : D
##### General Discussion
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Re: If x^9 = 9^9^9, what is the value of x?  [#permalink]

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30 Jul 2019, 08:33
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1
suchita2409 wrote:
x^9 = 9^9^9
x=9^9
B

Posted from my mobile device

Replacing x by 9^9 would mean (9^9)^9= 9^81

See that the number in question is much larger.
(9^9^8)^9= 9^9^8*9= 9^9^9

Intern
Joined: 21 Feb 2018
Posts: 29
If x^9 = 9^9^9, what is the value of x?  [#permalink]

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30 Jul 2019, 20:29
vinayakvaish wrote:
suchita2409 wrote:
x^9 = 9^9^9
x=9^9
B

Posted from my mobile device

Replacing x by 9^9 would mean (9^9)^9= 9^81

See that the number in question is much larger.
(9^9^8)^9= 9^9^8*9= 9^9^9

Am unable to understand this explanation. Can someone explain it once again. Tia.
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Re: If x^9 = 9^9^9, what is the value of x?  [#permalink]

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05 Aug 2019, 18:14
Bunuel wrote:
If $$x^9 = 9^{9^9}$$, what is the value of $$x$$?

A. $$9^8$$

B. $$9^9$$

C. $$9^{8^9}$$

D. $$9^{9^8}$$

E. $$9^{9^9}$$

Note that 9^9^9 is NOT the same as (9^9)^9 = 9^81. It is, instead, 9^(9^9), which is a much greater number than (9^9)^9.

Let’s take the 9th root of each side or rather, let’s raise each side to the power of 1/9:

(x^9)^(1/9) = [9^(9^9)]^(1/9)

x = 9^((9^9)*(1/9))

Since 1/9 = 9^(-1), we have:

x = 9^((9^9)*(9^(-1)))

x = 9^(9^(9 + (-1)))

x = 9^(9^8)

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Intern
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Posts: 22
Re: If x^9 = 9^9^9, what is the value of x?  [#permalink]

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05 Aug 2019, 19:49
Hello Can someone clarify one point for me please ?
Does :
(x^a)^b = x^(a*b) ?

x^a^b = x^(a*a*a*...*a) for example if b = 4 -> x^(a*a*a*a) ?

Thank you
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Re: If x^9 = 9^9^9, what is the value of x?  [#permalink]

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06 Aug 2019, 00:03
1
Coggi wrote:
Hello Can someone clarify one point for me please ?
Does :
(x^a)^b = x^(a*b) ?

x^a^b = x^(a*a*a*...*a) for example if b = 4 -> x^(a*a*a*a) ?

Thank you

(x^a)^b = x^(a*b)

But x^a^b is not = (x^a)^b

It starts from the top ... For example x^a^b^c^d^e will start from solving d^e then c^d^e then b^c^d^e and so on.

2^3^4 = 2^81

Whereas (2^3)^4 = 8^4 which is completely different.

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Joined: 20 Jul 2012
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GMAT 1: 650 Q47 V33
Re: If x^9 = 9^9^9, what is the value of x?  [#permalink]

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06 Aug 2019, 00:16
Solution is attached

Posted from my mobile device
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IMG_20190806_133921932.jpg [ 2.19 MiB | Viewed 795 times ]

Re: If x^9 = 9^9^9, what is the value of x?   [#permalink] 06 Aug 2019, 00:16