Question

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

(i) \(\frac{3}{8}\)

(ii) \(\frac{13}{125}\)

(iii) \(\frac{7}{80}\)

(iv) \(\frac{14588}{625}\)

(v) \(\frac{129}{(2^2 \times 5^7)}\)

Answer 1

(i) \(\frac{3}{8}\)

The given rational number is \(\frac{3}{8}\) 

It’s seen that, 8 = 2³ is of the form 2^m x 5ⁿ, where m = 3 and n = 0. 

So, the given number has terminating decimal expansion.

(ii) \(\frac{13}{125}\)

The given rational number is \(\frac{13}{125}\). 

It’s seen that, 125 = 5³ is of the form 2^m x 5ⁿ, where m = 0 and n = 3. 

So, the given number has terminating decimal expansion. 

∴ \(\frac{13}{125}\) 

= \(\frac{(13\, \times\, 23)}{(125\, \times\, 23)}\) 

= \(\frac{104}{1000}\)

= 0.104

(iii) \(\frac{7}{80}\)

The given rational number is \(\frac{7}{80}\). 

It’s seen that, 80 = 2⁴ x 5 is of the form 2^m x 5ⁿ, where m = 4 and n = 1. 

So, the given number has terminating decimal expansion.

(iv) \(\frac{14588}{625}\)

The given rational number is \(\frac{14588}{625}\). 

It’s seen that, 625 = 5⁴ is of the form 2^m x 5ⁿ, where m = 0 and n = 4. 

So, the given number has terminating decimal expansion.

(v) \(\frac{129}{(2^2 \times 5^7)}\)

The given number is \(\frac{129}{(2^2 \times 5^7)}\). 

It’s seen that, 2² x 5⁷ is of the form 2^m x 5ⁿ, where m = 2 and n = 7. 

So, the given number has terminating decimal expansion.