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★ Question asked by akshat ★​

Answer 1

$\huge \red{ \underline{ \underline { \sf \: Solution}}}$

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 1}}}}$

$\sf \sqrt{25} = \sqrt{ {5}^{2} }$

$\sf \implies \: 5$

$\implies \sf\frac{5}{1}$

Since, it is of the form p/q, It is a rational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 2}}}}$

We can write 43 as a square of some number. So, √43 is not a perfect square. Hence, it is an irrational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 3}}}}$

$\sf \sqrt{169 } = \sqrt{ {13}^{2} }$

$\sf \implies \: 13$

$\implies \sf \frac{13}{1}$

Since, it is the form of p/q, it is a rational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 4}}}}$

$\sf \sqrt{1.44} = \sqrt{1.2}$

$\sf \implies1.2$

$\implies \sf \frac{1.2}{1}$

Since, it is of the form p/q, it is a rational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 5}}}}$

We can write 1/3 as a square of some number. So, 1/√3 is not a perfect square. Hence, it is an irrational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 6}}}}$

It is a terminating decimal. Therefore, it is a rational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 7}}}}$

It is a irrational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 8}}}}$

It is equivalent to the fraction 8/10. Hence, it is a rational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 9}}}}$

It is a non - terminating. Therefore, it is a irrational number.

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$\huge \underline{ \boxed{ \sf{ \purple{solution \: 10}}}}$

$\sf2.356565656... = 2.3 \bar5 \bar6$

It is a non - terminating and repeating decimal. Therefore, it is a rational number.

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Answer 2

$\begin{array}{|c|c|}\cline{1-2} \sf \sqrt{25} & \sf Rational\\ & \\ \sf \sqrt{43} & \sf Irrational\\ & \\ \sf \sqrt{169} & \sf Rational\\ & \\ \sf \sqrt{1.44} & \sf Rational\\ & \\ \sf \dfrac{1}{\sqrt{3}} & \sf Irrational\\ & \\ \sf 3.2576 & \sf Rational\\ & \\ \sf 9\sqrt{2} & \sf Irrational\\ & \\ \sf 0.\overline{8} & \sf Rational\\ & \\ \sf 1.040040004 ... & \sf Irrational\\ & \\ \sf 2.356565656 ... & \sf Rational\\\cline{1-2}\end{array}$

Note :- View the above latex from website.

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Here, We need to find whether the given numbers are irrational or rational, So, Firstly we need to know these numbers can be expressed as p/q form or not, If they can be expressed they are rational & if they are not, They are irrational or the square root of that number must be a perfect square number.

General form of finding whether it's rational or irrational is :

→ If the decimal value is repeating is rational & if it's non-repeating it's irrational.

Hence,

Given numbers ::

• √25

• √43

• √169

• √1.44

• 1/√3

• 3.2576

• 9√2

• $\sf 0. \overline{8}$
• 1.040040004..

• 2.356565656...

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \large{ \sqrt{25} ::} \\ \\ : \implies \sf \dfrac{5}{1}$

• √25 is perfect square number.
• √25 can be expressed as a p/q form.
• It is a rational number.

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• √43 ::

→ 6.5574(Approx)

• Square root of 43 is a not a perfect square.
• √43 cannot be expressed as p/q form.
• It is a irrational number.

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• √169 ::

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \large{ \sqrt{169} ::} \\ \\ : \implies \sf \dfrac{13}{1}$

• √169 is perfect square number.
• √169 can be expressed as a p/q form.
• It is a rational number.

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \large{ \sqrt{1.44} ::} \\ \\ : \implies \sf 1.2 \: \: \: or \: \: \: \dfrac{6}{5}$

• √1.44 is not perfect square number.
• √1.44 can be expressed as a p/q form.
• It is a rational number.

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• 1/√3

→ 0.5773502691896(Approx)

• 1/√3 is not perfect square number.
• 1/√3 cannot be expressed as a p/q form.
• It is a irrational number.

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• 3.2576

→ 2036/625

• 3.2576 is not perfect square number.
• 3.2576 can be expressed as a p/q form.
• It is a rational number.

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• 9√2

→ 12.727922061357(Approx)

• 9√2 is not perfect square number.
• 9√2 cannot be expressed as a p/q form.
• It is a irrational number.

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• $\sf 0. \overline{8}$

→ 8/9

• 0.8 bar is not perfect square number.
• 0.8 bar can be expressed as a p/q form.
• It is a rational number.

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• 1.040040004..

→ 26001000/250000000

• 1.040040004 is not perfect square number.
• 1.040040004 can be expressed as a p/q form.
• 1.040040004.. is non-Terminating.
• It is a Irrational number.

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• 2.356565656...

→ 294570707/125000000

• 2.356565656 is not perfect square number.
• 2.356565656 can be expressed as a p/q form.
• It is a rational number.

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More to know ::

• When a number can be expressed as a ratio or p/q form it is called as rational number.

• If a number cannot be expressed as a ratio i.e p/q form it is called as irrational number.

For Example :- √23, 9√2 etc.

• A number is said to be as rational number if it is a non-terminating, repeating decimal.

★ General Form of irrational number ★

➤ The decimal number is non-terminating, It do not have an end. This kind of number cannot be solved if we use them as ratios.

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