**Denominator of the Rational Number 987/10500 in the Form 2^m.5^n**

To write the denominator of the rational number 987/10500 in the form 2^m.5^n, we need to determine the prime factors of the denominator and express it as a product of powers of 2 and 5.

Step 1: Prime Factorization of the Denominator
We begin by finding the prime factors of 10500. We can start by dividing 10500 by 2 repeatedly until we can no longer divide evenly.

10500 ÷ 2 = 5250
5250 ÷ 2 = 2625
2625 ÷ 2 = 1312.5 (not divisible by 2)

So, we have factored out as many 2's as possible. Now, let's determine the prime factors.

1312.5 ÷ 5 = 262.5 (not divisible by 5)
262.5 ÷ 5 = 52.5 (not divisible by 5)

Again, we have factored out as many 5's as possible. Let's continue with the prime factorization.

52.5 ÷ 2 = 26.25 (not divisible by 2)
26.25 ÷ 5 = 5.25 (not divisible by 5)

We have factored out all the prime factors, and the remaining number (5.25) cannot be divided evenly by either 2 or 5.

Step 2: Expressing the Prime Factors as Powers of 2 and 5
From the prime factorization, we can see that 10500 can be expressed as 2^2 × 5^2 × 262.5. Since 262.5 cannot be simplified further, we can rewrite the denominator as:

10500 = 2^2 × 5^2 × 262.5

Step 3: Writing the Denominator in the Form 2^m.5^n
To express the denominator in the required form, we need to rewrite 262.5 as a product of powers of 2 and 5.

262.5 ÷ 2 = 131.25 (not divisible by 2)
131.25 ÷ 5 = 26.25 (not divisible by 5)

We cannot simplify 262.5 any further. Therefore, the denominator can be expressed as:

10500 = 2^2 × 5^2 × 262.5 = 2^2 × 5^2 × 262.5

So, the denominator of the rational number 987/10500 can be written in the form 2^m.5^n as 2^2 × 5^2 × 262.5.

**Decimal Expansion of the Rational Number 987/10500 without Actual Division**
To write the decimal expansion of the rational number 987/10500 without actual division, we can use the fact that any number divided by a power of 10 can be written by shifting the decimal point to the left by the number of zeros in the power.

In this case, the denominator 10500 can be expressed as 2^2 × 5^2 × 262.5. We have two powers of 10, which means we need to shift the decimal point two places to the left.

987

987/10500 is not in the the form of p/q where pa, q are coprime.987/10500=329/3500=329/2^2*5^3*7. Hence it has non terminating decimal expansion.