Write the Decimal Expansions of the Given Rational Numbers

Video Explanation

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Solution

Question: Write down the decimal expansions of the following rational numbers by writing their denominators in the form of 2^m × 5ⁿ, where m and n are non-negative integers.

Important Rule

If the denominator of a rational number in its lowest form can be expressed as 2^m × 5ⁿ, then its decimal expansion is terminating.

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(i) 3/8

8 = 2³

∴ 3/8 = 0.375

Decimal expansion: 0.375

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(ii) 13/125

125 = 5³

13/125 = 0.104

Decimal expansion: 0.104

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(iii) 7/80

80 = 2⁴ × 5

7/80 = 0.0875

Decimal expansion: 0.0875

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(iv) 14588/625

625 = 5⁴

14588/625 = (14588 × 16) / (625 × 16)

= 233408 / 10000

= 23.3408

Decimal expansion: 23.3408

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(v) 129 / (22 × 57)

22 = 2 × 11
57 = 3 × 19

So, denominator = 2 × 3 × 11 × 19

The denominator contains prime factors other than 2 and 5.

∴ This rational number cannot be written in the form 2^m × 5ⁿ.

Hence, its decimal expansion is non-terminating repeating.

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Final Answer

(i) 0.375
(ii) 0.104
(iii) 0.0875
(iv) 23.3408
(v) Non-terminating repeating decimal

Conclusion

Thus, by expressing the denominators in the form 2^m × 5ⁿ, we can easily determine and write the decimal expansions of the given rational numbers.