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

Denominator of the rational number 257 5000 is 5000 . Now, factors of 5000 = 2 × 2 × 2 × 5 × 5 × 5 × 5 = ( 2 ) 3 × ( 5 ) 4 , which is of the type 2 m × 5 n , where m = 3 and n = 4 are non-negative integers. ∴ Rational number = 257 5000 = 257 2 3 × 5 4 × 2 2 [since, multiplying numerator and denominator by 2 ] 514 2 4 × 5 4 = 514 10 4 514 10000 = 0 . 0514 Hence, which is the required decimal expansion of the rational 257 5000 and it is also a terminating decimal number.

Prove that p + q is irrational, where p and q are primes.

Let us suppose that p + q is rational. Again, let p + q = a , where a is rational. Therefore, q = a − p On squaring both sides, we get q = a 2 + p − 2 a p [ ∵ ( a − b ) 2 = a 2 + b 2 − 2 a b ] Therefore, p = a 2 + p - q 2 a , which is a contradiction as the right hand side is rational number while p is irrational, since p and q are prime number. Hence, p + q is irrational.

E M B I B E

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 11

MEDIUM

10th CBSE

IMPORTANT

Earn 100

Show that $12^{n}$ cannot end with the digit $0$ or $5$ for any natural number $n$ .

Important Questions on Real Numbers

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 12

On a morning walk, three persons step off together and their steps measure

40 cm, 42 cm

and

45 cm

, respectively. What is the minimum distance each should walk, so that each can cover the same distance in complete steps?

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 13

Write the denominator of rational $\frac{257}{5000}$ in the form $2^{m} \times 5^{n}$ , where $m, n$ are non-negative integers. Hence, write its decimal expansion, without actual division.