Write the Denominator of 257/5000 in the Form 2^m × 5ⁿ and Find Its Decimal Expansion

Video Explanation

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Solution

Question: Write the denominator of the rational number 257/5000 in the form 2^m × 5ⁿ, where m and n are non-negative integers. Hence, write the decimal expansion without actual division.

Step 1: Prime Factorisation of the Denominator

5000 = 5 × 1000

1000 = 10³ = (2 × 5)³

∴ 5000 = 2³ × 5⁴

Thus, the denominator is in the form 2^m × 5ⁿ.

Step 2: Write the Decimal Expansion

Since the denominator has only the prime factors 2 and 5, the decimal expansion is terminating.

Make the denominator a power of 10:

5000 × 2 = 10000

∴ Multiply numerator and denominator by 2:

257/5000 = (257 × 2) / (5000 × 2)

= 514 / 10000

= 0.0514

Final Answer

Denominator = 2³ × 5⁴
Decimal expansion of 257/5000 = 0.0514

Conclusion

Thus, by expressing the denominator in the form 2^m × 5ⁿ, we can write the decimal expansion of the given rational number without performing long division.