Write the denominator of the rational number \(\dfrac{257}{5000}\) in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.
Denominator: \(2^3 \cdot 5^4\). Decimal: 0.0514
Step 1: Prime factorisation of the denominator.
We start with the denominator \(5000\).
Break it down: \(5000 = 5 \times 1000\).
Now, \(1000 = 10^3 = (2 \cdot 5)^3 = 2^3 \cdot 5^3\).
So, \(5000 = 5 \times (2^3 \cdot 5^3) = 2^3 \cdot 5^4\).
Thus, in the required form, the denominator is \(2^3 \cdot 5^4\).
Step 2: Find the number of decimal places.
The highest power among \(2^3\) and \(5^4\) is \(5^4\).
This means the decimal expansion will terminate after 4 places.
Step 3: Write the decimal without division.
The fraction is \(\dfrac{257}{5000}\).
We can think of this as dividing numerator and denominator in steps.
First, divide \(257\) by \(5\):
\(\dfrac{257}{5} = 51.4\).
Now we still need to divide by \(1000\) (since \(5000 = 5 \times 1000\)).
So, \(\dfrac{257}{5000} = \dfrac{51.4}{1000} = 0.0514\).
Final Answer:
The denominator in prime form is \(2^3 \cdot 5^4\). The decimal expansion is \(0.0514\).