The value of `[(1xx3xx9+2xx6xx18+3xx9xx27+.....)/(1xx5xx25+2xx10xx50+3xx15xx75+...)]^(1//3)` is

Options

A. `2/5`
B. `5/7`
C. `3/5`
D. `7/9`

Answer

To solve the problem, we need to evaluate the expression:

\[ \left[\frac{1 \times 3 \times 9 + 2 \times 6 \times 18 + 3 \times 9 \times 27 + \ldots}{1 \times 5 \times 25 + 2 \times 10 \times 50 + 3 \times 15 \times 75 + \ldots}\right]^{\frac{1}{3}} \]

### Step 1: Identify the Pattern in the Numerator
The terms in the numerator can be expressed as follows:
- The first term is \(1 \times 3 \times 9\)
- The second term is \(2 \times 6 \times 18\)
- The third term is \(3 \times 9 \times 27\)

We can see that:
- The first term can be rewritten as \(1 \times (3 \times 3)\)
- The second term can be rewritten as \(2 \times (3 \times 6)\)
- The third term can be rewritten as \(3 \times (3 \times 9)\)

Thus, we can factor out \(3^3\) from each term:
\[ = 3^3 \left(1 \times 1 + 2 \times 2 + 3 \times 3 + \ldots\right) \]

### Step 2: Identify the Pattern in the Denominator
The terms in the denominator can be expressed as follows:
- The first term is \(1 \times 5 \times 25\)
- The second term is \(2 \times 10 \times 50\)
- The third term is \(3 \times 15 \times 75\)

We can see that:
- The first term can be rewritten as \(1 \times (5 \times 5)\)
- The second term can be rewritten as \(2 \times (5 \times 10)\)
- The third term can be rewritten as \(3 \times (5 \times 15)\)

Thus, we can factor out \(5^3\) from each term:
\[ = 5^3 \left(1 \times 1 + 2 \times 2 + 3 \times 3 + \ldots\right) \]

### Step 3: Simplify the Expression
Now we can rewrite the entire expression:
\[ \frac{3^3 \left(1^2 + 2^2 + 3^2 + \ldots\right)}{5^3 \left(1^2 + 2^2 + 3^2 + \ldots\right)} \]

The \(1^2 + 2^2 + 3^2 + \ldots\) terms cancel out:
\[ = \frac{3^3}{5^3} \]

### Step 4: Calculate the Final Value
Now we take the cube root of the simplified expression:
\[ \left(\frac{3^3}{5^3}\right)^{\frac{1}{3}} = \frac{3}{5} \]

### Final Answer
Thus, the value of the original expression is:
\[ \frac{3}{5} \]