Question

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2ᵐ × 5ⁿ, where m, n are non-negative integers.
(i) 3/8 (ii) 13/125
(iii) 7/80 (iv) 14588/625
(v) 129/(2² × 5⁷)

Answer 1

Concept :

If the factors of denominator of the given rational number is of form  2ᵐ × 5ⁿ ,where n and m are non negative integers, then the decimal expansion of the rational number is terminating otherwise non terminating recurring.

 

SOLUTION :  

(i) Given : 3/8

We can write it as : 3/ 2³

Here, the factors of the denominator 8 are 2³× 5^0, which is in the form 2ᵐ × 5ⁿ .

So , 23/8 has terminating  decimal expansion.

The decimal expansion of 3/ 8 :  

⅜ = 3×5³ / 2³ × 5³

= 3 × 125 / (2×5)³  

= 375 / 10³ = 375 /1000  

= 0.375  

Hence, the decimal expansion of 3/8 is 0.375.

 

(ii) Given : 13/125

We can write it as :13/5³

Here, the factors of the denominator 125 are 2⁰ × 5³ which is in the form 2ᵐ × 5ⁿ .

So , 13/125 has a terminating  decimal expansion.

The decimal expansion of 13/ 125 :  

13/125 = 13 ×2³ /5³× 2³

= 13 × 8 /(5×2)³

= 104 / 10³ = 104/1000  

= 0.104  

Hence, the decimal expansion of 13/125 is 0.104.

 

(iii) Given : 7/80

Here, the factors of the denominator 80 are 2⁴ × 5¹, which is in the form 2ᵐ × 5ⁿ .

So , 7/80 has  a terminating decimal expansion.

The decimal expansion of 7/80 :  

7/80 = 7 × 5³ / 2⁴ × 5¹ × 5³

= 7 × 125 / (2×5)⁴

= 875/10⁴ = 875/10000

= 0.0875  

Hence, the decimal expansion of 7/80 is 0.0875 .

 

(iv) Given : 14588/625

Here, the factors of the denominator 625 is 2⁰ × 5⁴, which is in the form 2ᵐ 5ⁿ

So , 14588/625 has a terminating decimal expansion.

The decimal expansion of 14588/625 :  

14588/625 = 14588 × 2⁴ / 5⁴× 2⁴

= 14588 × 16 / (5×2)⁴

= 233408/ 10⁴

= 23.3408

Hence, the decimal expansion of 14588/625 is 23.3408.

 

(v) Given : 129/(2² × 5⁷)

129/(2² × 5⁷)

Here, the factors of the denominator  are 2² ×  5⁷  which is in the form 2ᵐ × 5ⁿ .

So ,129/2² × 5⁷  has a terminating decimal expansion.

The decimal expansion of 129/2² × 5⁷ :  

129 × 2⁵ /2² ×  5⁷ × 2⁵

= 129 × 32 / (2 × 5)⁷

= 4128 / 10000000

= 0.0004128

Hence, the decimal expansion of 129/(2² × 5⁷) is 0.0004128.

HOPE THIS ANSWER WILL HELP YOU…

 

Some more questions :  

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

(i) $\frac{23}{8}$

(ii) $\frac{125}{441}$

(iii) $\frac{35}{50} [NCERT]$

(iv) $\frac{77}{210} [NCERT]$

(v) $\frac{129}{2^{2} * 5^{7} * 7^{17}}$

(vi) $\frac{987}{10500}$

https://brainly.in/question/6751559

 

​What can you say about the prime factorisations of the denominators of the following rationals:

(i) 43.123456789

(ii) $43.\overline{123456789}$

(iii) $27.\overline{142857}$

(iv) 0.120120012000120000 ...

https://brainly.in/question/9770131

Answer 2

see the attachment friend !!