Following real number has decimal expansion as given below. Decide whether rational, or not. If rational, and of the form $\frac{p}{q}$ , what can you say about the prime factors of q? 43.123456789

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Solution

43.123456789
Since, the decimal expansion terminates, so the given real number is rational and therefore form 43.123456789
= $\frac{43123456789}{1000000000}$
= $\frac{43123456789}{10^{9}}$
= $\frac{43123456789}{(2 \times 5)^{9}}$
= $\frac{43123456789}{2^{9} \times 5^{9}}$
Hence, $q = 2^{9} 5^{9}$
The prime factorization of q is of the form $2^{n} 5^{m}$ , where n=9, m=9.

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Following real number has decimal expansion as given below. Decide whether rational, or not. If rational, and of the form pq, what can you say about the prime factors of q? 43.123456789

43.123456789 Since, the decimal expansion terminates, so the given real number is rational and therefore form 43.123456789 =431234567891000000000 =43123456789109 =43123456789(2×5)9 =4312345678929×59 Hence, q=2959 The prime factorization of q is of the form 2n5m, where n=9, m=9.

Following real number has decimal expansion as given below. Decide whether rational, or not. If rational, and of the form $\frac{p}{q}$ , what can you say about the prime factors of q? 43.123456789