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Answer :

i.

\({13 \over {3125}} = {13 \over {5^5}} = {13 × 2^5 \over {5^5 × 2^5}} = {13 × 2^5 \over {10^5}} = {416 \over {10^5}} = 0.00416\)

ii.

\({17 \over {8}} = {17 \over {2^3}} = {17 × 5^3 \over {2^3 × 5^3}} = {17 × 5^3 \over {10^3}} = {2125 \over {10^3}} = 2.125\)

iii .

\({15 \over {1600}} = {15 \over {2^6 × 5^2}} = {15 × 5^4 \over {2^6 × 5^2 × 5^4}} = {15 × 5^4 \over {10^6}} = {9375 \over {10^6}} = 0.009375\)

iv.

\({23 \over {2^3 × 5^2}} = {23 × 5^1 \over {2^3 × 5^2 × 5^1}} = {115 \over {10^3}} = 0.115\)

v.\({6 \over {15}} = {2 × 3 \over {3 × 5}} = {2 \over {5}} = {2 × 2 \over {5 × 2}} = {4 \over {10}} = 0.4\)

vi .\({35 \over {50}} = {5 ×7 \over {2 ×5^2}} = {7 \over {2 × 5}} = {7 \over {10}} = 0.7 \)

- 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.\({13 \over {3125}}\)ii.\({17 \over {8}}\)iii.\({64 \over {455}}\)iv.\({15 \over {1600}}\)v. \({29 \over {343}}\)vi.\({23 \over {2^3 × 5^2}}\)vii.\({{129} \over{2^2 × 5^7 × 7^5}}\)viii.\({6 \over {15}}\)ix.\({35 \over {50}}\) x.\({77 \over {210}}\)
- The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of the form \(\frac{p}{q}\), what can you say about the prime factors of q?1) 43.1234567892) 0.120120012000120000…3) 43.123456789123456789...

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