In this chapter we will discuss questions related to identification of irrational numbers with detailed solution.

**Question 01**

Identify if the below number is irrational or not.

\mathtt{( a) \ \sqrt{13}}\\\ \\ \mathtt{( b) \ \sqrt{441}}\\\ \\ \mathtt{( c) \ \sqrt{25}}\\\ \\ \mathtt{( d) \ \sqrt{15}}\\\ \\ \mathtt{( e) \ \sqrt{26}}

**Solution**

\mathtt{( a) \ \sqrt{13}}

Number 13 is not a perfect square, hence it’s an** irrational number**.

\mathtt{( b) \ \sqrt{441}}

441 is a perfect square.

Square root of 441 is 21.

\mathtt{\sqrt{441} \ =\ 21}

Hence, it’s **not an irrational number**.

\mathtt{( c) \ \sqrt{25}}

Number 25 is a perfect square.

\mathtt{\sqrt{25} \ =\ 5}

Hence, it’s not an** irrational number**.

\mathtt{( d) \ \sqrt{15}}

15 is not a perfect square, hence it’s an **irrational number**.

\mathtt{( e) \ \sqrt{26}}

26 is not a perfect square, hence it’s an** irrational number**.

**Question 02**

Identify if the below given number is irrational or not.

\mathtt{( a) \ \sqrt[3]{8}}\\\ \\ \mathtt{( b) \ \sqrt[3]{36}}\\\ \\ \mathtt{( c) \ \sqrt{81}}\\\ \\ \mathtt{( d) \ \sqrt{29}}\\\ \\ \mathtt{( e) \ \sqrt[3]{125}}

**Solution**

\mathtt{( a) \ \sqrt[3]{8}}

Number 8 is a perfect cube.

Cube root is 8 is 2.

\mathtt{\sqrt[3]{8} \ =\ 2}

Hence, the given number is** not an irrational number**.

\mathtt{( b) \ \sqrt[3]{36}}

36 is not a perfect cube.

Hence, it’s an **irrational number**.

\mathtt{( c) \ \sqrt{81}}

Number 81 is a perfect square.

The square root of 81 is 9.

\mathtt{\sqrt{81} \ =\ 9}

Hence, it’s **not an irrational number**.

\mathtt{( d) \ \sqrt{29}}

29 is not a perfect square.

Hence, the given number is** irrational number**.

\mathtt{( e) \ \sqrt[3]{125}}

Number 125 is a perfect cube.

Cube root of 125 is 5.

\mathtt{\ \sqrt[3]{125} =5}

Hence, the given number is **not irrational number**.

**Question 03**

Check if the value of variable is rational or irrational number.

\mathtt{( a) \ x^{2} =\ 144}\\\ \\ \mathtt{( b) \ y^{2} =0.9}\\\ \\ \mathtt{( c) \ z^{2} ={\frac{16}{361}}}\\\ \\ \mathtt{( d) \ t^{2} =0.36}\\\ \\ \mathtt{( e) \ n^{2} =7}

**Solution**

\mathtt{( a) \ x^{2} =\ 144}\\\ \\ \mathtt{x=\sqrt{144}}\\\ \\ \mathtt{x=\ 12}

Hence, x is **not an irrational number**.

(b) \mathtt{y=\sqrt{0.9}}\\\ \\ \mathtt{y\ =\ \sqrt{\frac{9}{10}}}\\\ \\ \mathtt{y=\frac{3}{\sqrt{10}}}

Since, the denominator is not a perfect square, the given fraction is an **irrational number**.

\mathtt{( c) \ z^{2} =\frac{16}{361}}\\\ \\ \mathtt{z =\sqrt{\frac{16}{361}}}\\\ \\ \mathtt{z=\frac{4}{19}}

Hence, the given number is **not an irrational number**.

\mathtt{( d) \ t^{2} =0.36}\\\ \\ \mathtt{t=\sqrt{0.36}}\\\ \\ \mathtt{t\ =\ \sqrt{\frac{36}{100}}}\\\ \\ \mathtt{t\ =\ \frac{6}{10}}

The given number is **not an irrational number**.

\mathtt{( e) \ n^{2} =7}\\\ \\ \mathtt{n=\sqrt{7}}

Since n is not a perfect square, the variable “n” is an **irrational number**

**Question 04**

Check if the below decimal is irrational or not.

(a) 1.6363636363 . . .

(b) 10.026011457 . . .

(c) 0.97125125125125 . . .

(d) 75.8

(e) 115.97052011 . . .

**Solution**

**(a) 1.6363636363 . . . **

Note that number 63 is repeated again & again.

The above decimal is non terminating and repeating, hence it’s** not an irrational number**.

**(b) 10.026011457 . . .**

This decimal number is non terminating and non repeating, so **it’s an irrational number**.

**(c) 0.97125125125125 . . .**

Note the number 125 after decimal point is repeated again and again.

Since the decimal is non terminating and repeating, it’s **not an irrational number.**

**(d) 75.8**

This is terminating decimal, so it’s **not an irrational number.**

**(e) 115.97052011 . . . **

It’s an non terminating and non repeating decimal. Hence, it’s an** irrational number**.

**Next chapter :** **Finding irrational number between two given numbers**