Question

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m x 5ⁿ, where m, n are 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) \(\frac{3}{8}\) = \(\frac{3}{2^3}\) = \(\frac{3\times 5^3}{2^3\times 5^3}\) the denominator is in the form of 2^m x 5ⁿ

⇒ \(\frac{375}{10^3}\) = 0.375

(ii) \(\frac{13}{125}\)=\(\frac{13}{5^3}\) = \(\frac{13\times 2^3}{2^3\times 5^3}\) the denominator is in the form of 2^m x 5ⁿ

⇒ \(\frac{104}{10^3}\) = 0.104

(iii) \(\frac{7}{80}\) = \(\frac{7}{2^4\times 5}\) = \(\frac{7\times 5^3}{2^4\times 5^4}\) the denominator is in the form of 2^m x 5ⁿ

⇒ \(\frac{875}{10^4}\) = 0.0875

(iv) \(\frac{14588}{625}\) = \(\frac{22\times 7\times521}{5^4}\) = \(\frac{22\times 2^4\times 7\times521}{2^4\times 5^4}\) the denominator is in the form of 2^m x 5ⁿ

⇒ \(\frac{22\times 2^4\times 7\times521}{(2\times 5)^4}\) = 23.3408

(v) \(\frac{129}{2^2\times 5^7}\) = \(\frac{2^5\times 129}{2^2\times 5^7}\) = \(\frac{2^5\times 129}{(2\times 5)^7}\) the denominator is in the form of 2^m x 5ⁿ

⇒ \(\frac{2^5\times 129}{(2\times 5)^7}\) = 0.0004128