Mathematics
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
Rational Irrational Nos
Answer
The given number is in its lowest form.
Prime factorization of denominator 3125:
3125 = 5 x 5 x 5 x 5 x 5 x 1
= 55 x 1
= 1 x 55
= 20 x 55 [∵ 20 = 1]
Denominator is of the form 2m x 5n, where m, n are non-negative integers.
∴ The given number has a terminating decimal expansion.
The given number is in its lowest form.
Prime factorization of denominator 8:
8 = 2 x 2 x 2 x 1
= 23 x 1
= 23 x 50 [∵ 50 = 1]
Denominator is of the form 2m x 5n, where m, n are non-negative integers.
∴ The given number has a terminating decimal expansion.
The given number is in its lowest form.
Prime factorization of denominator 75:
75 = 3 x 5 x 5 x 1
= 3 x 52 x 1
= 3 x 52 x 20 [∵ 20 = 1]
Denominator has a prime factor 3 other than 2 or 5.
∴ The given number has a non-terminating repeating decimal expansion.
Both numerator and denominator contain common factor 3. Reducing the number to its lowest form:
The Denominator 5 = 20 x 51
Denominator is of the form 2m x 5n, where m, n are non-negative integers.
∴ The given number has a terminating decimal expansion.
The given number is in its lowest form.
Prime factorization of denominator 625:
625 = 5 x 5 x 5 x 5 x 1
= 54 x 1
= 1 x 54
= 20 x 54 [∵ 20 = 1]
Denominator is of the form 2m x 5n, where m, n are non-negative integers.
∴ The given number has a terminating decimal expansion.
Both numerator and denominator contain common factor 7. Reducing the number to its lowest form:
Prime factorization of denominator 30:
30 = 2 x 3 x 5
Denominator has a prime factor 3 other than 2 or 5.
∴ The given number has a non-terminating repeating decimal expansion.
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