The number of decimal places after which the decimal expansion of the rational number `9/(2^4 times 5)` will terminate is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of decimal places after which the decimal expansion of the rational number \( \frac{9}{2^4 \times 5} \) will terminate, we can follow these steps:

### Step 1: Identify the form of the rational number
The given rational number is \( \frac{9}{2^4 \times 5} \).

### Step 2: Determine the factors of the denominator
The denominator \( 2^4 \times 5 \) can be rewritten as \( 2^4 \times 5^1 \).

### Step 3: Make the denominator a power of 10
To determine how many decimal places the decimal will terminate, we need to express the denominator in the form \( 10^n \). The number \( 10 \) can be factored into \( 2 \times 5 \).

### Step 4: Equalize the powers of 2 and 5
To convert \( 2^4 \times 5^1 \) into \( 10^n \), we need to equalize the powers of \( 2 \) and \( 5 \). The smaller power between \( 2^4 \) and \( 5^1 \) is \( 5^1 \). We need to make the power of \( 5 \) equal to \( 4 \) by multiplying and dividing by \( 5^{3} \).

### Step 5: Rewrite the expression
We can multiply and divide the numerator and denominator by \( 5^3 \):
\[ \frac{9 \times 5^3}{2^4 \times 5^4} \]

### Step 6: Calculate the numerator
Calculating the numerator:
\[ 9 \times 5^3 = 9 \times 125 = 1125 \]

### Step 7: Calculate the denominator
Calculating the denominator:
\[ 2^4 \times 5^4 = 16 \times 625 = 10000 \]

### Step 8: Form the new rational number
Now we have:
\[ \frac{1125}{10000} \]

### Step 9: Convert to decimal
Dividing \( 1125 \) by \( 10000 \):
\[ \frac{1125}{10000} = 0.1125 \]

### Step 10: Count the decimal places
The decimal \( 0.1125 \) has 4 decimal places.

### Final Answer
The number of decimal places after which the decimal expansion of the rational number \( \frac{9}{2^4 \times 5} \) will terminate is **4**.

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