Without actual division show that each of the following rational numbers is a non-terminating repeating decimal 129+2^2 x 5^7 x 7^5

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Solution

Dear Student,

$F o r a r a t i o n a l n u m b e r t o h a v e a t e r m i n a t i n g d e c i m a l p o i n t, t h e p r i m e f a c t o r i s a t i o n o f t h e d e n o m i n a t o r m u s t o n l y c o n t a i n p o w e r s o f 2 a n d 5 . T h a t i s t h e d e n o m i n a t o r m u s t c o n t a i n o n l y 2^{x} \times 5^{y} w h e r e x a n d y a r e a n y i n t e g e r s . I n t h e r a t i o n a l n u m b e r \frac{129}{2^{2} \times 5^{7} \times 7^{5}} t h e d e n o m i n a t o r c o n t a i n s p o w e r s o f 7 i n a d d i t i o n t o t h e p o w e r s o f 2 a n d 5 . T h e r e f o r e t h e r a t i o n a l n u m b e r \frac{129}{2^{2} \times 5^{7} \times 7^{5}} h a s a n o n t e r m i n a t i n g d e c i m a l p o i n t .$

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Similar questions

Question 1 (vii)
Without actually performing the long division, state whether $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

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

(i)

\frac{11}{(2^{3} \times 3)}

(ii)

\frac{73}{(2^{2} \times 3^{3} \times 5)}

(iii)

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

(iv)

\frac{9}{35}

(v)

\frac{77}{210}

(vi)

\frac{32}{147}

(vii)

\frac{29}{343}

(viii)

\frac{64}{455}

Q. Without actually performing the long division, 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}

(iv)

\frac{77}{210}

(v)

\frac{129}{2^{2} \times 5^{7} \times 7^{17}}

Question 1 (vii)
Without actually performing the long division, state whether $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

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

\frac{129}{2^{2} 5^{7} 7^{5}}

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