**Question 1. 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)13/3125 (ii)17/8 (iii)64/455

(iv)15/1600

(v)29/343

(vi)23/2³*5²

(vii)129/2²* 5^{7}* 7^{5}

(viii)6/15

(ix)35/50

(x)77/210

**Answer:**

(i) 13/3125

Factorize the denominator we get

3125 =5 x 5 x 5 x 5 x 5 = 5^{5}

So denominator is in form of 5^{m} so 13/3125 is terminating .

(ii) 17/8

Factorize the denominator we get

8 =2 x 2 x 2 = 2³

So denominator is in form of 2^{n} so 17/8 is terminating .

(iii) 64/455

Factorize the denominator we get

455 =5 x 7 x 13

There are 7 and 13 also in denominator so denominator is not in form of 2^{n}*5^{m} . hence 64/455 is not terminating.

(iv) 15/1600

Factorize the denominator we get

1600 =2 x 2 x 2 x2 x 2 x 2 x 5 x 5 = 2^{6} x 5²

so denominator is in form of 2^{n} x 5^{m}

Hence 15/1600 is terminating.

(v) 29/343

Factorize the denominator we get

343 = 7 x 7 x 7 = 7³

There are 7 also in denominator so denominator is not in form of 2^{n} x 5^{m}

Hence it is none - terminating.

(vi) 23/(2³ x 5²)

Denominator is in form of 2^{n} x 5^{m}

Hence it is terminating.

(vii) 129/(2² x 5^{7} x 7^{5} )

Denominator has 7 in denominator so denominator is not in form of 2^{n} x 5^{n}

Hence it is none terminating.

(viii) 6/15

divide nominator and denominator both by 3 we get 2/5

Denominator is in form of 5^{m} so it is terminating.

(ix) 35/50 divide denominator and nominator both by 5 we get 7/10

Factorize the denominator we get

10=2 x 5

So denominator is in form of 2^{n} x5^{m} so it is terminating

(x) 77/210.

simplify it by dividing nominator and denominator both by 7 we get 11/30

Factorize the denominator we get

30=2 x 3 x 5

Denominator has 3 also in denominator so denominator is not in form of 2^{n} x 5^{n}

Hence it is none terminating.