Without actual division show that each of the following rational numbers is a nonterminating repeating decimal. (i) 11/2^3 x 3

Question

Without actual division show that each of the following rational numbers is a nonterminating repeating decimal.

(i) \(\frac{11}{2^3\times3}\) 

(ii) \(\frac{73}{2^3\times3^3\times5}\)

(iii) \(\frac{129}{2^2\times5^7\times7^5}\)

(iv) \(\frac{9}{35}\)

Answer 1

 (i) \(\frac{11}{2^3\times3}\)  

We know either 2 or 3 is not a factor of 11, so it is in its simplest form. 

Moreover, (2³× 3) ≠ (2^m × 5ⁿ) 

Hence, the given rational is non – terminating repeating decimal.

(ii) \(\frac{73}{2^3\times3^3\times5}\) 

We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form. 

Moreover, (2²× 3³ ×5) ≠ (2^m × 5ⁿ) 

Hence, the given rational is non-terminating repeating decimal

(iii) \(\frac{129}{2^2\times5^7\times7^5}\) 

We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form. 

Moreover, (2² × 5⁷ × 7⁵ ) ≠ (2^m × 5ⁿ)

Hence, the given rational is non-terminating repeating decimal

(iv) \(\frac{9}{35}\) = \(\frac{9}{5\times7}\)

We know either 5 or 7 is not a factor of 9, so it is in its simplest form. 

Moreover, (5 × 7) ≠ (2^m × 5ⁿ) 

Hence, the given rational is non-terminating repeating decimal.