Write the denominator of the rational number 257/5000 in the form 2^m × 5^n, where m, n are non-negative integers. Hence, write its decimal expansion without actual division.

The given number is in its lowest form. Prime factorization of denominator 5000: 5000 = 2 x 2 x 2 x 5 x 5 x 5 x 5 = 2 3 x 5 4 Hence, denominator of rational number in the form 2 m × 5 n is 2 3 x 5 4 where m = 3 and n = 4.

Mathematics

Write the denominator of the rational number $\dfrac{257}{5000}$ in the form 2^m × 5ⁿ where m, n are non-negative integers. Hence, write its decimal expansion without actual division.

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Answer

The given number $\dfrac{257}{5000}$ is in its lowest form.

Prime factorization of denominator 5000:

$\begin{array}{l|l} 2 & 5000 \ \hline 2 & 2500 \ \hline 2 & 1250 \ \hline 5 & 625 \ \hline 5 & 125 \ \hline 5 & 25 \ \hline 5 & 5 \ \hline & 1 \end{array}$

5000 = 2 x 2 x 2 x 5 x 5 x 5 x 5
= 2³ x 5⁴

Hence, denominator of rational number $\dfrac{257}{5000}$ in the form 2^m × 5ⁿ is 2³ x 5⁴ where m = 3 and n = 4.

$\dfrac{257}{5000} = \dfrac{257}{2^3 \times 5^4} \\[0.5em] = \dfrac{257 \times 2}{2^3 \times 5^4 \times 2} \\[0.5em] = \dfrac{514}{2^4 \times 5^4} \\[0.5em] = \dfrac{514}{(2 \times 5)^4} \\[0.5em] = \dfrac{514}{10^4} \\[0.5em] = 0.0514 \\[0.5em] \bold{\therefore \dfrac{257}{5000} = 0.0514}$

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Mathematics

Write the denominator of the rational number $\dfrac{257}{5000}$ in the form 2^m × 5ⁿ where m, n are non-negative integers. Hence, write its decimal expansion without actual division.

Rational Irrational Nos

Answer

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