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

1. 9^8
2. 9^9
3. 9^8^9
4. 9^9^8
5. 9^9^9

Correct Answer: D
Solution and Explanation:
Approach Solution 1:
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)

Approach Solution 2:
Taking the ninth root of the equation or taking power = 1/9 of the equation.

Given

(x^9)1/9= (9^9^9)1/9
*Property: (a^m)n = a^m.n
Equation: x^9.1/9= 9^9^9.1/9

Cancelling 9&9 on L.H.S
x^1= 9^9^9/9
*Property: a^m/a^n= a^m-n
Equation: x= 9^9^9-1= 9^9^8
Therefore value of x is 9^9^8

Approach Solution 3:
x^9= 9^9^9
x= (9^9^9)1/9
= 9^9^9*9-1
= 9^9(9-1)
= 9^9^8

Now if we do x^9
9^9^9.9^9^8….9^9^8
9 times
(9^9^8)^9
= 9^9^8.9
= 9^9(8+1)
= 9^9^9 which verifies x= 9^9^8

“If x^9= 9^9^9, what is the value of x?”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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