Question

1.The decimal representation of 137/2³ 5⁴ 7² is : (a) terminating decimal (b) non-terminating decimal (c) non-terminating non-repeating decimal (d) none of these​

Answer 1

Answer:

We know that the divisor of the forms 2n5m always form a terminating decimal number. Let us simplify the expressions and find if the expansion have a terminating decimal expansion. 

Option A : 

21077

=2×3×5×77×11

=2×3×511

Since there is a factor of 3 in the denominator, the decimal expansion will not be terminating.

Option B : 

3023

=2×3×523

Since, the denominator contains a power of 3 , it is non-terminating. 

Option C : 

441125

=3×3×7×75×5×5

This is also non-terminating. 

Option D : 

823

=2×2×223

This contains power of 2 in the denominator. Hence, the decimal expansion is terminating.

Answer 2

Given:

The decimal representation of

$\frac{137}{2^{3} \times 5^{4} \times 7^{2}}$

Theorem state

• $\text { Let } x=\frac{p}{q} \text { be a rational number, }$

• $\text { the prime factorization of } \mathrm{q} \text { is not of the form } 2^{n} \times 5^{m}$

• $\text { where m and } \mathrm{n} \text { are }\text { non-negative integers. }$

• $\text { Then } x \text { has a decimal expression which is non terminating }$ repeating
• It is clear that the prime factorization of denominator is not of the form

        $2^{n} \times 5^{m}$

It has terminating decimal expansion

So the correct option is "a"

Hence it is terminating decimal expansion