Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal :
`9/80`

Answer

To determine whether the rational number \( \frac{9}{80} \) is expressible as a terminating decimal without performing actual division, we can follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator, which is 80.

\[ 80 = 8 \times 10 \]

Now, we can further factor these numbers:

\[ 8 = 2^3 \quad \text{and} \quad 10 = 2 \times 5 \]

So, we can combine these factors:

\[ 80 = 2^3 \times (2 \times 5) = 2^4 \times 5^1 \]

### Step 2: Check the Form of the Denominator
Next, we need to check if the denominator can be expressed in the form of \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

From our factorization, we have:

\[ 80 = 2^4 \times 5^1 \]

This shows that the denominator is indeed in the required form.

### Step 3: Conclusion
Since the denominator \( 80 \) can be expressed as \( 2^4 \times 5^1 \), we conclude that the rational number \( \frac{9}{80} \) is expressible as a terminating decimal.

Thus, \( \frac{9}{80} \) is a terminating decimal.

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