Tag: rational and irrational numbers

Quizzes (6) Questions (185)

Questions Related to rational and irrational numbers

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

The smallest of $\sqrt[3]{4},    \sqrt[4]{5},     \sqrt[4]{6},    \sqrt[3]{8}$ is:

  1. $\sqrt[3]{8}$
  2. $\sqrt[4]{5}$
  3. $\sqrt[3]{4}$
  4. $\sqrt[4]{6}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(B) $\sqrt[3]{4}, \sqrt[4]{5},  \sqrt[4]{6}, \sqrt[3]{8}$

$=4^{1/3}, 5^{1/4}, 6^{1/4}, 8^{1/3}$

L.C.M of 3 & 4 $=12$

So, the given surds can be written as,

$=4^{4/12}, 5^{3/12}, 6^{3/12}, 8^{4/12}$

$=(4^{4})^{1/12}, (5^{3})^{1/12}, (6^{3})^{1/12}, (8^{4})^{1/12}$

$=(256)^{1/12}, (125)^{1/12}, (216)^{1/12}, (4096)^{1/12}$

$\therefore $ The smallest one is $\sqrt[4]{5}$

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

$\sqrt{11}-\sqrt{10} .... \sqrt{12}-\sqrt{11}$,use appropriate inequality to fill the gap.

  1. <

  2. >
  3. $=$

  4. cannot determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We first consider $\sqrt { 11 } -\sqrt { 10 }$ as follows:


$\sqrt { 11 } -\sqrt { 10 } =3.317-3.162=0.156$

Now we find the value of $\sqrt { 12 } -\sqrt { 11 }$ as follows:

$\sqrt { 12 } -\sqrt { 11 } =3.464-3.317=0.147$


Since $0.156>0.147$

Hence, $\sqrt { 11 } -\sqrt { 10 }>\sqrt {12} -\sqrt {11}$

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

If $p=\sqrt{32}-\sqrt{24}$ and $q=\sqrt{50}-\sqrt{48}$

  1. $p< q$
  2. $p> q$
  3. $p=q$
  4. $p\leq q$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(B) $\frac{p}{q}=\frac{\sqrt{32}-\sqrt{24}}{\sqrt{50}-\sqrt{48}}\times \frac{\sqrt{50}+\sqrt{48}}{\sqrt{50}+\sqrt{48}}$

$=\frac{(4\sqrt{2}-2\sqrt{6})(5\sqrt{2}+4\sqrt{3})}{2}$

$=(2\sqrt{2}-\sqrt{6})(5\sqrt{2}+4\sqrt{3})> 1$

$\therefore p> q$

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

If $x=\sqrt{2}+1,    y=\sqrt{17}-\sqrt{2}$, then:

  1. $x< y$
  2. $x > y$
  3. $x=y$
  4. $x\geq y$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(A) Given, $x=\sqrt{2}+1$ and $y=\sqrt{17}-\sqrt{2}$

$\dfrac{x}{y}=\dfrac{\sqrt{2}+1}{\sqrt{17}-\sqrt{2}}\times \dfrac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}$

$=\dfrac{(\sqrt{2}+1)(\sqrt{17}+\sqrt{2})}{17-2}$

$=\dfrac{(\sqrt{2}+1)(\sqrt{17}+\sqrt{2})}{15}< 1$

$\therefore x< y$

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

Arrange the following in ascending order of magnitude: $\displaystyle \sqrt[4]{90}, \sqrt[3]{10}, \sqrt{6}$

  1. $\displaystyle \sqrt{3} < \sqrt[4]{10} < \sqrt[3]{6}$
  2. $\displaystyle \sqrt{3} > \sqrt[4]{10} > \sqrt[3]{6}$
  3. $\displaystyle \sqrt{3} > \sqrt[4]{10} < \sqrt[3]{6}$
  4. $\displaystyle \sqrt{3} < \sqrt[4]{10} > \sqrt[3]{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

$if\,A\, = \sqrt 7  - \sqrt 6 \,and\,B = \,\sqrt 6  - \sqrt {5,} \,then\,$

  1. $A > B$
  2. $A = B$
  3. $A < B\,$
  4. $A \geqslant B$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Given numbers are $A=\sqrt{7}-\sqrt{6}$ and $B=\sqrt{6}- \sqrt{5}$
Let $x =\sqrt{5}$ and $y= \sqrt{7}$ then by
$A.M$ and $G.M$
$\dfrac{x+y}{2} \le \sqrt{\dfrac{x^{2}+y^{2}}{2}}$
$\Rightarrow \dfrac{\sqrt{5}+ \sqrt{7}}{2} \le \sqrt{6}$
$\Rightarrow \sqrt{5}+ \sqrt{7} \le 2 \sqrt{6}$
$\Rightarrow \sqrt{7}- \sqrt{6} \le \sqrt{6} - \sqrt{5}$
Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Which of the following numbers is the least ?
$\displaystyle (0.5)^{2},\sqrt{0.49},\sqrt[3]{0.008},0.23$

  1. $\displaystyle (0.5)^{2}$
  2. $\displaystyle \sqrt{0.49}$
  3. $\displaystyle \sqrt[3]{0.008}$

  4. 0.23
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ (0.5)^{2}=0.25$
$\sqrt{0.49}=0.7;$
$ \sqrt[3]{0.008}=\sqrt[3]{.2^3}=0.2$
$0.23$
Arranging in ascending order the numbers are $0.2< 0.23< 0.25< 0.7$
$ \therefore \sqrt[3]{0.008}=0.2$ is the least

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

The greatest number among $\displaystyle \sqrt[3]{2},\sqrt{3},\sqrt[3]{5}$ and $1.5$ is 

  1. $\displaystyle \sqrt[3]{2}$
  2. $\displaystyle \sqrt{3}$
  3. $\displaystyle \sqrt[3]{5}$
  4. $1.5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

LCM of $3, 2 = 6$
Given numbers are $ \sqrt[3]{2},\sqrt{3},\sqrt[3]{5}, 1.5$ i.e,
$ 2^{1/3},3^{1/2},5^{1/3},1.5$
$ \therefore $ Raising each number to power $6$, we get
$ (2^{1/3})^{6},(3^{1/2})^{6},(5^{1/3})^{6}, (1.5)^{6}$

$= 2^{2},3^{3},5^{2}, \left(\cfrac{3}{2}\right)^{6}$
$=4,27,25,\cfrac{729}{64}$
Of all these numbers, $27$ is the greatest.
$ \Rightarrow \sqrt{3}$ is the greatest. 

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

The smallest of $\displaystyle \sqrt{8}+\sqrt{5},\sqrt{7}+\sqrt{6},\sqrt{10}+\sqrt{3}$ and $\displaystyle \sqrt{11}+\sqrt{2}$ is 

  1. $\displaystyle \sqrt{8}+\sqrt{5}$
  2. $\displaystyle \sqrt{7}+\sqrt{6}$
  3. $\displaystyle \sqrt{10}+\sqrt{3}$
  4. $\displaystyle \sqrt{11}+\sqrt{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \sqrt{8}+\sqrt{5}=2.83+2.24=5.07$
$\displaystyle \sqrt{7}+\sqrt{6}=2.65+2.45=5.09$
$\displaystyle \sqrt{10}+\sqrt{13}=3.16+3.61=6.77$
$\displaystyle \sqrt{11}+\sqrt{12}=3.32+1.41=4.73$
$\displaystyle \therefore $ Smallest is $\displaystyle \sqrt{11}+\sqrt{2}$