Question

please help me with this question with explanation​

Answer 1

Answer:

(4) this is a answer

$\frac{2}{5} \frac{79}{80} \: \: \: \: \: \: 0.9913730875$

this is a answer I hope this is a correct answer ok

Answer 2

$\large\underline{\sf{Given \:Question - }}$

Which of the following irrational numbers is smallest ?

$\rm :\longmapsto\: \: \sqrt[100]{\dfrac{1}{4} }, \: \sqrt[200]{\dfrac{1}{8} }, \: \sqrt[300]{\dfrac{1}{32} }, \: \sqrt[400]{\dfrac{1}{64} }$

$\: \: \: \rm \: (1). \: \sqrt[300]{\dfrac{1}{32} }$

$\: \: \: \rm \: (2). \: \sqrt[400]{\dfrac{1}{64} }$

$\: \: \: \rm \: (3). \: \sqrt[200]{\dfrac{1}{8} }$

$\: \: \: \rm \: (4). \: \sqrt[100]{\dfrac{1}{4} }$

$\large\underline{\sf{Solution-}}$

Given irrational numbers are

$\rm :\longmapsto\: \: \sqrt[100]{\dfrac{1}{4} }, \: \sqrt[200]{\dfrac{1}{8} }, \: \sqrt[300]{\dfrac{1}{32} }, \: \sqrt[400]{\dfrac{1}{64} }$

Let we first convert the indices of each irrational number same.

So, we have to first find the LCM of 100, 200, 300 and 400

So, using Prime factorization method,

$\red{\rm :\longmapsto\:Prime \: Factors \: of \: 100 = {2}^{2} \times {5}^{2}}$

$\red{\rm :\longmapsto\:Prime \: Factors \: of \: 200 = {2}^{3} \times {5}^{2}}$

$\red{\rm :\longmapsto\:Prime \: Factors \: of \: 300 = {2}^{2} \times3 \times {5}^{2}}$

$\red{\rm :\longmapsto\:Prime \: Factors \: of \: 400 = {2}^{4} \times {5}^{2}}$

So,

$\bf\implies \:LCM [ 100, 200,300,400 ] = {2}^{4} \times 3 \times {5}^{2} = 1200$

We know,

$\green{\boxed{ \bf{ \: \: \: \sqrt[y]{x} = \sqrt[yz]{ {x}^{z} } \: \: \: }}}$

So, the given irrational numbers can be rewritten as

Consider,

$\rm :\longmapsto\:\sqrt[100]{\dfrac{1}{4} }$

can be rewritten as

$\rm \: = \: \sqrt[100 \times 12]{\dfrac{1}{ {4}^{12} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ {( {2}^{2} )}^{12} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ {2}^{24} } }$

Now, Consider

$\rm :\longmapsto\:\sqrt[200]{\dfrac{1}{8} }$

can be rewritten as

$\rm \: = \: \sqrt[200 \times 6]{\dfrac{1}{ {8}^{6} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ { {(2}^{3} )}^{6} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ {2}^{18} } }$

Now, Consider

$\rm :\longmapsto\:\sqrt[300]{\dfrac{1}{32} }$

can be rewritten as

$\rm \: = \: \sqrt[300 \times 4]{\dfrac{1}{ {32}^{4} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ {( {2}^{5} )}^{4} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ {2}^{20} } }$

Now, Consider

$\rm :\longmapsto\:\sqrt[400]{\dfrac{1}{64} }$

can be rewritten as

$\rm \: = \: \sqrt[400 \times 3]{\dfrac{1}{ {64}^{3} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ { {(2}^{6} )}^{3} } }$

$\rm \: = \: \sqrt[1200]{\dfrac{1}{ {2}^{18} } }$

So,

$\bf\implies \:\sqrt[1200]{\dfrac{1}{ {2}^{24} } } < \sqrt[1200]{\dfrac{1}{ {2}^{20} } } < \sqrt[1200]{\dfrac{1}{ {2}^{18} } }$

$\bf\implies \:\sqrt[100]{\dfrac{1}{4} } < \sqrt[300]{\dfrac{1}{32} } < \sqrt[200]{\dfrac{1}{8} } = \sqrt[400]{\dfrac{1}{64} }$

Hence,

$\bf\implies \:Smallest \: irrational \: number \: is \: \sqrt[100]{\dfrac{1}{4} }$

• So, option (4) is correct