What is the value of `9^(1//3)X 9^(1//9)X 9^(1//27)`. . . . `oo` ?
(a)9
(b)3
(c)1
(d)`9^(1//3)`

Options

A. 9
B. 3
C. `9^(1//3)`
D. 1

Answer

To solve the problem \( 9^{1/3} \times 9^{1/9} \times 9^{1/27} \times \ldots \) up to infinity, we can follow these steps:

### Step 1: Identify the series
The expression can be rewritten as:
\[ 9^{1/3} \times 9^{1/9} \times 9^{1/27} \times \ldots \]
Since the bases are the same (all are 9), we can combine the exponents.

### Step 2: Combine the exponents
Using the property of exponents that states \( a^m \times a^n = a^{m+n} \), we can add the exponents:
\[ 9^{(1/3) + (1/9) + (1/27) + \ldots} \]

### Step 3: Identify the series of exponents
The series of exponents is:
\[ \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots \]
This is a geometric series where the first term \( a = \frac{1}{3} \) and the common ratio \( r = \frac{1/9}{1/3} = \frac{1}{3} \).

### Step 4: Sum the infinite geometric series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values we have:
\[ S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]

### Step 5: Substitute the sum back into the exponent
Now we substitute the sum back into the exponent:
\[ 9^{(1/3) + (1/9) + (1/27) + \ldots} = 9^{1/2} \]

### Step 6: Simplify the expression
We can simplify \( 9^{1/2} \):
\[ 9^{1/2} = \sqrt{9} = 3 \]

### Final Answer
Thus, the value of \( 9^{1/3} \times 9^{1/9} \times 9^{1/27} \times \ldots \) is:
\[ \boxed{3} \]