Without actually performing the long division , state whether the following rational no. will have a terminating decimal expansion or non-te

Question

Without actually performing the long division , state whether the following rational no. will have a terminating decimal expansion or non-terminating repeating decimal expansion.
(a)6/15​

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Eliza 1 month 2021-09-16T16:43:54+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-09-16T16:45:11+00:00

    SOLUTION:

    \frac{6}{15} = \frac{2 \times \not{3}}{\not{3} \times 5} = \frac{2}{5}

    Here, prime factors of 5 = 5¹ × 2⁰ which is in form 2^{n}\times 5^{m}

    So, it will have a terminating decimal expansion.

    .

    \underline{\sf \red{More}\ \blue{Information}}

    Here is a theorem on the decimal expansion of a rational number.

    Let ‘x’ be a rational number whose decimal expansion terminates.

    Then, ‘x’ can be expressed in the form (p/q), where ‘p’ and ‘q’ are coprime and the prime factorization of ‘q’ is in the form 2^{n}\times 5^{m}, where n and m are non-integers.

    Here is the decimal expansion of  \frac{6}{15}

    \frac{6}{15} = \frac{2 \times \bf{ 2^1}}{5^{1} \times \bf 2^1} = \frac{4}{10} = 0.4

    It terminates after one decimal place.

    0
    2021-09-16T16:45:31+00:00

    To check it whether 6/15 is a terminating decimal place or not.

    We will so prime factorization,

    6 = 2*3

    and 15 = 3*5

    3 will be cancelled

    the fraction becomes simple

    2/5 here 5 is the denominator which is in the form 2^n * 5^m, where 2^0 is there.

    SO it is a terminating decimal place

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