Question

Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal:

(viii)$\frac{129}{2^2\times 5^7\times 7^5}$

Answer 1

Classifying the number as non-terminating repeating decimal:

When the denominator has a factor other than the factor of the form $2^m\times 5^n$(where m and n are positive integers), then the rational number is a non-terminating repeating decimal.

$\frac{129}{2^2\times 5^7\times 7^5}$$=\frac{3\times 43}{2^2\times 5^7\times 7^5}$

As the denominator has the factor $7^5$other than the factor of the form $2^m\times 5^n$.

Final answer: Hence, $\frac{\mathbf{129}}{{\mathbf{2}}^{\mathbf{2}}\mathbf{\times }{\mathbf{5}}^{\mathbf{7}}\mathbf{\times }{\mathbf{7}}^{\mathbf{5}}}$is a non-terminating repeating decimal.