Tag: rational and irrational numbers

Questions Related to rational and irrational numbers

State true or false 

$\sqrt { 7 } -\sqrt { 3 } $ is greater than $\sqrt { 5 } -1$ 

comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

  1. True

  2. False

Show answer
Correct Option: B

Which is greater?
${ \left( \cfrac { 1 }{ 2 }  \right)  }^{ 1/2 } $ or ${ \left( \cfrac { 2 }{ 3 }  \right)  }^{ 1/3 } $

comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

  1. ${ \left( \cfrac { 2 }{ 3 }  \right)  }^{ 1/3 } $

  2. ${ \left( \cfrac { 1 }{ 2 }  \right)  }^{ 1/2 } $

  3. Both are equal

  4. None of the above

Show answer
Correct Option: A
Explanation:

$ \left(\dfrac{1}{2}\right)^{1/2} \; or \; \left(\dfrac{2}{3}\right)^{1/3}$


$= \left(\left(\dfrac{1}{2}\right)^{1/2}\right)^6 \; or \; \left(\left(\dfrac{2}{3}\right)^{1/3}\right)^6$


$= \left(\dfrac{1}{2}\right)^3 \; or \; \left(\dfrac{2}{3}\right)^2$


$= \left(\dfrac{1}{8}\right) \; or \; \left(\dfrac{4}{9}\right)$


= 0.125 or 0.44


Since, 0.44 is greater and so is $ \left(\dfrac{2}{3}\right)^{1/3}$

The correct descending order of the following surds is 

$ \sqrt [3]{2}$, $\sqrt 3$, $\sqrt 4$, $\sqrt 5$

comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

  1. $\sqrt 3$ > $\sqrt 4$ > $\sqrt 5$ > $\sqrt [3]{2}$

  2. $\sqrt 4$ > $\sqrt 5$ > $\sqrt [3]{2}$ > $\sqrt 3$

  3. $\sqrt 5$ > $\sqrt 4$ > $\sqrt 3$ > $\sqrt [3]{2}$

  4. $\sqrt [3]{2}$ > $\sqrt 4$ > $\sqrt 3$ > $\sqrt 5$

Show answer
Correct Option: C
Explanation:

LCM of $3$ and $2$ is $6$.
Therefore, multiplying the index of all numbers by $6$.
${\sqrt [3]{2}}^6 = 2^\dfrac 63 = 2^2 = 4$

${\sqrt 3}^6 = 3^\dfrac 62 = 3^3 = 27$

${\sqrt 4}^6 = 4^ \dfrac 62 = 4^3 = 64$

${\sqrt 5}^6 = 5^{\dfrac 62} =  5^3 = 125$

$\therefore 125 > 64 > 27 > 4$

So, option $C$ is correct.

Which one of the following is an irrational number?

comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

  1. $\sqrt[3]{-27}$

  2. $\sqrt{2}(3\sqrt{2}+2\sqrt{8})$

  3. $\dfrac{3\sqrt{18}}{2\sqrt{6}}$

  4. $\sqrt{\dfrac{1}{2}}\cdot\sqrt{\dfrac{25}{2}}$

  5. $\dfrac{2\sqrt{5}}{\sqrt{45}}$

Show answer
Correct Option: C
Explanation:

Option A: $\sqrt [ 3 ]{ -27 } ={ (-3) }^{ 3\times \frac { 1 }{ 3 }  }=-3$
Option B: $\sqrt { 2 } (3\sqrt { 2 } +2\sqrt { 8 } )=\sqrt { 2 } (3\sqrt { 2 } +4\sqrt { 2 } )=\sqrt { 2 } (7\sqrt { 2 } )=14$
Option C: $\dfrac { 3\sqrt { 18 }  }{ 2\sqrt { 6 }  } =\dfrac { 3\sqrt { 3 }  }{ 2 } $
Option D: $\sqrt { \dfrac { 1 }{ 2 }  } \sqrt { \dfrac { 25 }{ 2 }  } =\dfrac { 5 }{ 2 } $
Option E: $\dfrac { 2\sqrt { 5 }  }{ \sqrt { 45 }  } =\dfrac { 2\sqrt { 5 }  }{ 3\sqrt { 5 }  } =\dfrac { 2 }{ 3 } $
Therefore, all are rational except option $C$.