If the prime factorization of a natural number n is 2 3 × 3 2 × 5 2 × 7 , write the number of consecutive zeroes in n .

It is given that, n = 2 3 × 3 2 × 5 2 × 7 To find the number of zeroes in the above, we need to identify the equal powers of 2 and 5 . ⇒ n = ( 7 × 9 × 2 ) × ( 2 2 × 5 2 ) ⇒ n = ( 7 × 9 × 2 ) × 100 ⇒ n = 12600 Hence, the number of consecutive zeroes are 2 .

Write the condition to be satisfied by q so that a rational number p q has a terminating decimal expansion.

We need to find the condition to be satisfied by q so that a rational number p q has a terminating decimal expansion. For the terminating decimal expansion, we should have a multiple of 10 in the denominator. Hence, the prime factorization of q must be of the form 2 m × 5 n , where m and n are non-negative integers.

Write the condition to be satisfied by q so that a rational number p q has a non-terminating decimal expansion.

We need to find the condition to be satisfied by q so that a rational number p q has a non-terminating decimal expansion. For the terminating decimal expansion, we should have a multiple of 10 in the denominator. Hence, for p q to have a non-terminating decimal expansion, the prime factorization of q is not of the form 2 m × 5 n , where m and n are non-negative integers.

The decimal expansion of the rational number 43 2 4 × 5 3 will terminate after how many places of decimals?

We have, 43 2 4 × 5 3 Let x = p q be a rational number, such that the prime factorization of q is of the form 2 m × 5 n , where m and n are non-negative integers. Then, x has a decimal expansion which terminates after k places, where k is the larger of m and n . This is clear that the prime factorization of the denominator is of the form 2 m × 5 n , where m = 4 and n = 3 , where 4 is larger than 3 . Hence, it has terminating decimal expansion which terminates after 4 places of decimal.

Has the rational number 441 2 2 × 5 7 × 7 2 a terminating or a non-terminating decimal representation?

We have, 441 2 2 × 5 7 × 7 2 On solving we get, 441 2 2 × 5 7 × 7 2 = 441 2 2 × 5 7 × 49 = 9 2 2 × 5 7 Let x = p q be a rational number, such that the prime factorization of q is of the form 2 m × 5 n , where m and n are non-negative integers, then the expansion is terminating. Here, it is clear that the prime factorization of the denominator is of the form 2 m × 5 n . Hence, it has terminating decimal expansion. "Note: Answer mentioned in the book is incorrect the correct answer is 'terminating'."