#### (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 …………. .

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.

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
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

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.

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

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.
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.

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.

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.

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.