Correct Answer – (i) 0.115 (ii) 0.192 (iii) 0.21375 (iv) 0.009375 (v) 0.053125 (vi) 0.00608
`(i)23/((2^(3)xx5^(2)))=(23xx5)/((2^(3)xx5^(3)))=115/((10)^(3))= 115/1000= 0.115`
`(ii) 24/124 = 24/(5^(3))xx 2^(3)/2^(3) = (24xx8)/((5xx2)^(3))= 192/((10)^(3))= 192/1000= 0.192`
(iii) ` 171/800 = 171/16 xx1/100 = 21.375/100 = 0.21375`
`(iv) 15/1600 = 15/16xx1/100 = (0.9375)/100 = 0.009375`
`(v) 17/320 = ( 17xx5)/(320xx5) = 85/16 xx1/100 = (5.3125)/100 = 0.053125`
`(vi) 19/3125 = (19xx8)/(3125xx8) = 152/25000 = 152/25 xx 1/1000 = 6.08/1000 = 0.00608`

129/2² x 5⁷ x 7⁵

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

Moreover, (2 ²× 5 ⁷× 7 ⁵ ) ≠ (2^m × 5ⁿ ) 

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

9/35 = 9/5 x 7

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.

(i) \(\frac{23}{2^3\times5^2}\) = \(\frac{23\times5}{2^3\times5^3}\) = \(\frac{115}{1000}\) = 0.115

We know either 2 or 5 is not a factor of 23, so it is in its simplest form 

Moreover, it is in the form of (2^m × 5ⁿ). 

Hence, the given rational is terminating

(ii) \(\frac{24}{125}\) = \(\frac{24}{5^3}\) = \(\frac{24\times2^3}{5^3\times2^3}\) = \(\frac{192}{1000}\) = 0.192

We know 5 is not a factor of 23, so it is in its simplest form. 

Moreover, it is in the form of (2^m × 5ⁿ ). 

Hence, the given rational is terminating.

(iii) \(\frac{171}{800}\) = \(\frac{171}{2^5\times5^2}\) = \(\frac{171\times5^3}{2^5\times5^5}\) = \(\frac{21375}{100000}\) = 0.21375

We know either 2 or 5 is not a factor of 171, so it is in its simplest form.

Moreover, it is in the form of (2^m × 5ⁿ ). 

Hence, the given rational is terminating.

(iv) \(\frac{15}{1600}\) = \(\frac{15}{2^6\times5^2}\) = \(\frac{15\times5^4}{2^6\times5^6}\) = \(\frac{9375}{1000000}\) = 0.009375

We know either 2 or 5 is not a factor of 15, so it is in its simplest form. 

Moreover, it is in the form of (2^m × 5ⁿ). 

Hence, the given rational is terminating.

(v) \(\frac{17}{320}\) = \(\frac{17}{2^6\times5}\) = \(\frac{17\times5^5}{2^6\times5^6}\) = \(\frac{53125}{1000000}\) = 0.053125

We know either 2 or 5 is not a factor of 17, so it is in its simplest form. 

Moreover, it is in the form of (2^m × 5ⁿ). 

Hence, the given rational is terminating

(vi) \(\frac{19}{3125}\) = \(\frac{19}{5^5}\) = \(\frac{19\times2^5}{5^5\times2^5}\) = \(\frac{608}{100000}\) = 0.00608

We know either 2 or 5 is not a factor of 19, so it is in its simplest form. 

Moreover, it is in the form of (2^m × 5ⁿ). 

Hence, the given rational is terminating.

11/2³ x 3

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.