Question

1. (3+$\sqrt{23}$ ) - $\sqrt{23}$
2. 2 $\sqrt{7}$ / 2 $\sqrt{7}$
3. 1/$\sqrt{2}$
4. 2$\pi$
irrational or rational answer each of them ;-;

Answer 1

Answer:

Topic

• Rational and Irrational

Solution

$\sf{1. \: (3+\sqrt{23}) - \sqrt{23}} \\ \\ \sf{ \implies3 + \sqrt{23} - \sqrt{23} } \\ \\ \sf{ \implies 3 + 0} \\ \\ \sf{ \implies \red{ 3}}$

⚘Therefore it is Rational.

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$\sf{2. \: \: \frac{ 2 \sqrt{7}} { 2 \sqrt{7}}}$

⇝Since numerator and denominator is same they will get cancel out.

$\sf{ \implies \red{1}}$

Therefore it is Rational.

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$3. \: \: \sf{ \frac{1}{\sqrt{2}}}$

⚘It is a Irrational number.

Explanation:

$\sf{Let \: us \: assume \: that \: \frac{1}{√2} \: can \: be \: expressed \: in \: \frac{a}{b} \: Form.}$

⇝Where a and b are co -prime number.

$\sf{ \implies\frac{1}{√2} = \frac{a}{b} } \\ \\ \sf{ \implies b = \sqrt{2} \: a } \\ \\ \sf{ \implies \frac{b}{a} = \sqrt{2} }$

But we know that √2 is irrational.

Therefore it contradicts the fact that a/b is Rational.

$\sf{ \red{So \: \frac{1}{ \sqrt{2} } \: is \: irrational}}$

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$4. \: \: \sf{2 \times \pi}$

⚘Since π is irrational so 2π is also irrational.

⚘Rational number multiplied to irrational number is always irrational.

${\boxed{\sf{\pink{Rational× Irrational= Irrational}}}}$

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More to know!!

❁Rational number -

• A rational number is the one which can be represented in the form of A/B where B≠0

• They are terminating

❁Irrational number -

• irrational numbers are those that can written in decimals but not in the fraction form, cannot be written as the ratio of two integers.

• They are non- terminating

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