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  • (i)Classify the given number as rational or irrational with justification. Squareroot 196

(i) Classify the given number as rational or irrational with justification.\sqrt{196}.
(ii) Classify the given number as rational or irrational with justification 3\sqrt{18}
(iii) Classify the given number as rational or irrational with justification.\sqrt{\frac{9}{27}}
(iv) Classify the given number as rational or irrational with justification.\frac{\sqrt{28}}{\sqrt{343}}
(v) Classify the given number as rational or irrational with justification.-\sqrt{0\cdot 4}
(vi) Classify the given number as rational or irrational with justification.\frac{\sqrt{12}}{\sqrt{75}}
(vii) Classify the given number as rational or irrational with justification.0.5918
(viii) Classify the given number as rational or irrational with justification.\left ( 1+\sqrt{5} \right )-\left ( 4+\sqrt{5} \right )
(ix) Classify the given number as rational or irrational with justification.10.124124 ………..
(x) Classify the given number as rational or irrational with justification.1.010010001 ………….

 

 

 

 

 

 


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Answers (1)

(i)Answer.   [Rational]
Solution.        
We have,
\sqrt{196} = 14 = \frac{14}{1} which follows rule of rational number.

Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.

So, \sqrt{196} is a rational number.

(ii)Answer.     [Irrational]
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
We have,

3\sqrt{18}= 3\sqrt{9\times 2}
= 3\sqrt{9}\sqrt{2}
= 3\times 3\sqrt{2}= 9\sqrt{2}
So, it can be written in the form of \frac{p}{q} as \frac{9\sqrt{2}}{1} 

But we know that 9\sqrt{2} is irrational
(Irrational numbers are real numbers which cannot be represented as simple fractions.)
Hence, 3\sqrt{18} is an irrational number
(iii)Answer.          [Irrational]
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
We have,

\sqrt{\frac{9}{27}}= \sqrt{\frac{3\times 3}{3\times 3\times 3}}
= \sqrt{\frac{1}{3}}= \frac{\sqrt{1}}{\sqrt{3}}= \frac{1}{\sqrt{3}}
So this can be written in the form of \frac{p}{q} as \frac{1}{\sqrt{3}} but we can see that \sqrt{3} (denominator) is irrational.
(Irrational numbers are real numbers which cannot be represented as simple fractions.)
Hence \sqrt{\frac{9}{27}} is irrational

(iv)Answer.          [Rational]
Solution.        
We have,

\frac{\sqrt{28}}{\sqrt{343}}= \frac{\sqrt{4\times 7}}{\sqrt{49\times 7}}
= \frac{2\times \sqrt{7}}{7\times \sqrt{7}}= \frac{2}{7}

Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Hence \frac{\sqrt{28}}{\sqrt{343}} is a rational number.

(v)Answer.    [Irrational]
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
We have,
-\sqrt{0\cdot 4}= -\sqrt{\frac{4}{10}}
= -\sqrt{\frac{2}{5}}= -\frac{\sqrt{2}}{\sqrt{5}}
So, it can be written in the form of \frac{p}{q} as \frac{-\sqrt{2}}{\sqrt{5}}
But we know that both \sqrt{2},\sqrt{5} are irrational

(Irrational numbers are real numbers which cannot be represented as simple fractions.)

Hence, -\sqrt{0\cdot 4} is an irrational number


(vi)Answer.          [Rational]
Solution.        
We have,

\frac{\sqrt{12}}{\sqrt{75}}= \frac{\sqrt{4\times 3}}{\sqrt{25\times 3}}
= \frac{\sqrt{4}\sqrt{3}}{\sqrt{25}\sqrt{3}}= \frac{\sqrt{4}}{\sqrt{25}}
= \frac{2}{5}
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q
should be rational when the fraction is expressed in the simplest form.

So, \frac{\sqrt{12}}{\sqrt{75}} is a rational number.
(vii)Answer.          [Rational]
Solution.        
We have,

0\cdot 5918= \frac{0\cdot 5918\times 10000}{1\times 10000}
= \frac{5918}{10000}= \frac{2959}{5000}

Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.

Also we can see that 0.5918 is a terminating decimal number hence it must be rational.

So, 0.5918 is a rational number.

(viii)Answer.          [Rational]
Solution.        
We have,

\left ( 1+\sqrt{5} \right )-\left ( 4+\sqrt{5} \right )
= 1+\sqrt{5}-4-\sqrt{5}
= -3= \frac{-3}{1}
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
So, \left ( 1+\sqrt{5} \right )-\left ( 4+\sqrt{5} \right ) is a rational number.

(ix)Answer.       [Rational]
Solution.        

Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
All non-terminating recurring decimal numbers are rational numbers.
Non terminating Recurring decimals are those decimals which have a particular pattern/sequence that keeps on repeating
itself after the decimal point. They are also called repeating decimals.
Examples: 1/3 = 0.33333…, 4/11 = 0.363636…
Now, 10.124124 ………. is a decimal expansion which is a non-terminating recurring.
So, it is a rational number.

(x)Answer.          [Irrational]
Solution.        

Non terminating Recurring decimals are those decimals which have a particular pattern/sequence that keeps on repeating itself after the decimal point.
All non-terminating recurring decimal numbers are rational numbers.
Non terminating Non Recurring decimals are those decimals which do not have a particular pattern/sequence after the decimal point and it does not end.
All non-terminating non-recurring decimal numbers are irrational numbers.
1.010010001 ………. is non-terminating non-recurring decimal number, therefore it cannot be written in the form \frac{p}{q};q\neq 0,with p,q both as integers.

Thus, 1.010010001 ……….. is an irrational number.

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