Question 33

Write the denominator of the rational number $\frac{\mathrm{257/}}{5000}$ in the form $2^m\times 5^n$, where m, n and non-negative integers. Hence, write its decimal expansion without actual division.

Solution:

To solve the problem of expressing the denominator of the rational number $$\frac{\mathrm{257/}}{5000}$$ in the form $2^m\times 5^n$ and finding its decimal expansion, we can follow these steps:

Step 1: Factor the Denominator
We start with the denominator, which is $$5000$$. We need to factor it into its prime factors.

$$5000=5\times 1000$$
Next, we can factor $$1000$$:
$$1000=10\times 100=(2\times 5)\times (10\times 10)=(2\times 5)\times (2\times 5)\times (2\times 5)={23}^{}\times {53}^{}$$
Thus, we can rewrite $$5000$$:
$$5000=5\times (23\times 53)=23\times 54$$

Step 2: Identify $$m$$ and $$n$$
From the factorization, we can see that:
$$5000=23\times 54$$
This means:
– $$m=3$$
– $$n=4$$

Step 3: Write the Decimal Expansion
Now we need to find the decimal expansion of $$\frac{\mathrm{257/}}{5000}$$ without performing the actual division.

To do this, we can multiply both the numerator and the denominator by $$2$$ to make the denominator a power of $$10$$:
$$\frac{\mathrm{257/}}{5000}\times \frac{\mathrm{2/}}{2}=(\frac{257}{\times }=\frac{\mathrm{514/}}{10000}$$

Step 4: Determine the Decimal Value
The denominator $$10000$$ is $${10}^4$$, which means we can easily express $$\frac{\mathrm{514/}}{10000}$$ as a decimal:
$$\frac{\mathrm{514/}}{10000}=0.0514$$

Final Answer
Thus, the decimal expansion of $$\frac{\mathrm{257/}}{5000}$$ is:
$$0.0514$$