Q2 of 23 Page 1

If a is a positive rational number and n is a positive integer greater than 1,prove that a n is a rational number.


a is a positive rational number (Given)

The product of two rational numbers is always a rational number. Hence if a is a rational number then-


a 2 = a × a is also a rational number,
a 3 = a 2 × a is also a rational number,
a 4 = a 3 × a is also a rational number,

…
∴ a n = a n-1 × a is a rational number.

More from this chapter

All 23 →
1

Write the following rational numbers in decimal form:
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

3

Find three rational numbers lying between 0 and 0.1. Find twenty rationalnumbers between 0 and 0.1. Give a method to determine any number ofrational numbers between 0 and 0.1.

4

Complete the following:
(i) Every point on the number line corresponds to a _____________  numberwhich may be either  _____________ or _____________.
(ii) The decimal form of an irrational number is neither _____________  or_____________.
(iii) The decimal representation of the rational number https://gs-post-images.s3.amazonaws.com/user_files/63094/15039/images/extra-1_files/Image3058.gif is _____________.
(iv) 0 is  _____________  number. [Hint: a rational /an irrational]

5
Which of the following rational numbers have the terminating decimalrepresentation?
(i)3/5             (ii)7/20                   (iii)2/13
(iv) 27/40        (v)133/125            (vi)23/7
[Making use of the result that a rational number p/q where p and q have nocommon factor(s) will have a terminating representation if and only if theprime factors of q are 2's or 5's or both.