A real number `( 2^(2) xx 3^(2) xx 7^(2))/(2^(2) xx 5^(2) xx 3^(2) xx 7^(4))` will have ______

Options

A. Terminating decimal expansion
B. Non-terminating decimal expansion
C. Repeating decimal expansion
D. Both B and C

Answer

To solve the given expression \(\frac{2^2 \times 3^2 \times 7^2}{2^2 \times 5^2 \times 3^2 \times 7^4}\), we will simplify it step by step.

### Step 1: Write down the expression
We start with the expression:
\[ \frac{2^2 \times 3^2 \times 7^2}{2^2 \times 5^2 \times 3^2 \times 7^4} \]

### Step 2: Cancel common factors in the numerator and denominator
We can cancel \(2^2\) and \(3^2\) from both the numerator and the denominator:
\[ = \frac{7^2}{5^2 \times 7^4} \]

### Step 3: Simplify the expression further
Now, we can simplify \(7^2\) and \(7^4\):
\[ = \frac{1}{5^2 \times 7^{4-2}} = \frac{1}{5^2 \times 7^2} \]

### Step 4: Write the final simplified expression
The final expression is:
\[ \frac{1}{5^2 \times 7^2} \]

### Step 5: Analyze the form of the number
The expression \(\frac{1}{5^2 \times 7^2}\) can be expressed as:
\[ \frac{1}{25 \times 49} = \frac{1}{1225} \]

### Step 6: Determine the nature of the number
Since the denominator contains prime factors \(5\) and \(7\) (which are not \(2\) or \(5\)), the decimal representation of this fraction will be non-terminating and repeating.

### Conclusion
Thus, the real number \(\frac{2^2 \times 3^2 \times 7^2}{2^2 \times 5^2 \times 3^2 \times 7^4}\) will have a non-terminating repeating decimal representation.

### Final Answer
The answer is: **non-terminating repeating decimal**.

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