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

Question

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}

Hanzel101 · Accepted Answer

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 = 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

vinayakvaish · Answer

Please see that (a^m)^n= a^m*n
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

Answer D

urshi · Answer

Am unable to understand this explanation. Can someone explain it once again. Tia.

ScottTargetTestPrep · Answer

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) =  ^(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)
 
Answer: D

Coggi · Answer

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

amarsinha · Answer

(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.

Hope this clarifies your doubt.

amarsinha · Answer

Solution is attached

Posted from my mobile device

Regor60 · Answer

X^9 = 9^9^9

Using Log base 9:

9LogX = 9^9Log9 = 9^9

LogX = 9^9/9 = 9^8

X = 9^9^8

Posted from my mobile device

Oppenheimer1945 · Answer

x^9=9^(9)^(9)=(9^{(8+1)})^9 =>x^9=(9^(9)^8)^9 =>x=9^9^8

ZIX · Answer

Here's an easier approach:

(x)^9 = 9^{9^9}

Therefore x must be raised to some power. Let  x = (a^b) :
 (a^b)^9 = 9^{9^9}

Separate the exponents:
 b*9 = 9^9 
 9^8*9^1 = 9^9

So:
 (a^{9^1})^{9^8} = 9^{9^9}  --- 1
(OR)
 (a^{9^8})^{9^1} = 9^{9^9}  --- 2

From the question, we know that we have to separate 9 from x ( (x)^9 = 9^{9^9} ), so we go with eq 2:

(9^{9^8})^9 = 9^{9^9}

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards