Questions on identification of irrational numbers

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


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