After how many places, the decimal form of the number `(27)/(2^(3)5^(4)3^(2))` will terminate?

Options

A. one
B. two
C. three
D. four

Answer

To determine after how many places the decimal form of the number \( \frac{27}{2^3 \cdot 5^4 \cdot 3^2} \) will terminate, we can follow these steps:

### Step 1: Factor the numerator and denominator
We start with the expression:
\[ \frac{27}{2^3 \cdot 5^4 \cdot 3^2} \]
We know that \( 27 = 3^3 \). Thus, we can rewrite the expression as:
\[ \frac{3^3}{2^3 \cdot 5^4 \cdot 3^2} \]

### Step 2: Simplify the expression
Next, we can simplify the fraction by canceling out \( 3^2 \) from the numerator and denominator:
\[ \frac{3^{3-2}}{2^3 \cdot 5^4} = \frac{3^1}{2^3 \cdot 5^4} = \frac{3}{2^3 \cdot 5^4} \]

### Step 3: Rewrite the denominator
Now, we can express the denominator in a more manageable form:
\[ 2^3 \cdot 5^4 = 2^3 \cdot 5^3 \cdot 5^1 = (2 \cdot 5)^3 \cdot 5^1 = 10^3 \cdot 5^1 \]
Thus, we can rewrite our expression as:
\[ \frac{3}{10^3 \cdot 5^1} \]

### Step 4: Multiply by a suitable form of 1
To make the denominator a power of 10, we can multiply the numerator and denominator by \( 2 \):
\[ \frac{3 \cdot 2}{10^3 \cdot 5^1 \cdot 2} = \frac{6}{10^4} \]

### Step 5: Convert to decimal form
Now, we can express this as:
\[ \frac{6}{10000} = 0.0006 \]

### Step 6: Count the decimal places
The decimal \( 0.0006 \) has 4 decimal places.

### Conclusion
Thus, the decimal form of the number \( \frac{27}{2^3 \cdot 5^4 \cdot 3^2} \) will terminate after 4 decimal places.

**Final Answer: 4**

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