Tag: exponents

Quizzes (9) Questions (51)

Questions Related to exponents

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 $\displaystyle \sqrt{11}+\sqrt{2}$

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

Which one of the following set of surds is correct sequence of ascending order of their values?

  1. $\displaystyle \sqrt[4]{10},\sqrt[3]{6},\sqrt{3}$
  2. $\displaystyle \sqrt{3},\sqrt[4]{10},\sqrt[3]{6},$
  3. $\displaystyle \sqrt{3},\sqrt{10},\sqrt[3]{6},$
  4. $\displaystyle \sqrt[4]{10},\sqrt{3},\sqrt[3]{6}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt[4]{10},\sqrt[3]{6},\sqrt{3}$
The order of the given irrational numbers are 2,3,4.
LCM of (2,3,4)=12
Now convert each irrational number as of order 12
$\sqrt[4]{10}=\sqrt[12]{10^3}=\sqrt[12]{1000}$
$\sqrt[3]{6}=\sqrt[12]{6^4}=\sqrt[12]{1296}$
$\sqrt{3}=\sqrt[12]{3^6}=\sqrt[12]{729}$
Hence, ascending order$\sqrt{3}<\sqrt[4]{10}<\sqrt[3]{6}$

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

Which is the greatest out of the following ?

  1. $\displaystyle \sqrt[3]{1.728}$
  2. $\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}+1}$
  3. $\displaystyle \left ( \frac{1}{2} \right )^{-2}$
  4. $\displaystyle \frac{17}{8}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$\Rightarrow  \sqrt[3]{1.728}=1.2$

$\Rightarrow \cfrac{\sqrt{3}-1}{\sqrt{3}+1}=\cfrac{(\sqrt{3}-1)^{2}}{(\sqrt{3}+1)(\sqrt{3}-1)}=\cfrac{3+1-2\sqrt{3}}{3-1}$
$ =\cfrac{4-2\sqrt{3}}{2}=2-\sqrt{3}$
$ =2-1.732=0.268$

$\Rightarrow \left ( \cfrac{1}{2} \right )^{-2}=2^{2}=4$

$\Rightarrow \cfrac{17}{8}=2.2125$

$ \therefore \left ( \cfrac{1}{2} \right )^{-2}$ is the greatest.