RD Sharma 2020 solution class 9 chapter 2 Exponents of Real Numbers MCQS

MCQS

Page-2.28

Question 1:

The value of ${\left\{2-3 (2-3{)}^3\right\}}^3$is

(a) 5

(b) 125

(c) 1/5

(d) -125

Answer 1:

We have to find the value of. So,

The value of is 125
Hence the correct choice is

Question 2:

The value of x − y^x-y when x = 2 and y = −2 is

(a) 18
(b) −18
(c) 14
(d) −14

Answer 2:

Given
Here
By substituting in we get 

The value of is – 14
Hence the correct choice is .

Question 3:

The product of the square root of x with the cube root of x is
(a) cube root of the square root of x
(b) sixth root of the fifth power of x
(c) fifth root of the sixth power of x
(d) sixth root of x

Answer 3:

We have to find the product (say L) of the square root of x with the cube root of x is. So, 

$=x^{\frac{3+2}{6}}=x^{\frac{5}{6}}$
The product of the square root of x with the cube root of x is
Hence the correct alternative is

Page-2.29

Question 4:

The seventh root of x divided by the eighth root of x is
(a) x

(b) $\sqrt{x}$

(c) $\sqrt[56]{x}$

(d) $\frac{1}{\sqrt[56]{x}}$

Answer 4:

We have to find he seventh root of x divided by the eighth root of x, so let it be L. So, 


The seventh root of x divided by the eighth root of x is
Hence the correct choice is .

Question 5:

The square root of 64 divided by the cube root of 64 is

(a) 64
(b) 2
(c) $\frac{1}{2}$
(d) 64^2/3

Answer 5:

We have to find the value of . 
So,


The value of is
Hence the correct choice is .

Question 6:

Which of the following is (are) not equal to ${\left\{{\left(\frac{5}{6}\right)}^{1/5}\right\}}^{-1/6}$?
(a) ${\left\{{\left(\frac{5}{6}\right)}^{\frac{1}{5}}\right\}}^{-\frac{3}{6}}$

(b) $\frac{1}{{\left\{{\left(\frac{5}{6}\right)}^{1/5}\right\}}^{1/6}}$

(c) ${\left(\frac{6}{5}\right)}^{1/30}$

(d) ${\left(\frac{5}{6}\right)}^{-1/30}$

Answer 6:

We have to find the value of
So,


Hence the correct choice is .

Question 7:

When simplified $(x^{-1}+y^{-1}{)}^{-1}$ is equal to

(a) xy

(b) x+y

(c) $\frac{xy}{x+y}$

(d) $\frac{x+y}{xy}$

Answer 7:

We have to simplify
So,


The value of is
Hence the correct choice is .

Question 8:

If $8^{x+1}$ = 64 , what is the value of $3^{2x+1}$ ?

(a) 1
(b) 3
(c) 9
(d) 27

Answer 8:

We have to find the value of provided
So,

Equating the exponents we get


By substitute in we get 

The real value of is
Hence the correct choice is .

Question 9:

If (2³)² = 4^x, then 3^x =

(a) 3
(b) 6
(c) 9
(d) 27

Answer 9:

We have to find the value ofprovided
So,

By equating the exponents we get

By substituting in we get 

The value of is
Hence the correct choice is

Question 10:

If x^{-2 =} 64, then x^1/3+x⁰ =

(a) 2
(b) 3
(c) 3/2
(d) 2/3

Answer 10:

We have to find the value ofif
Consider,


Multiply on both sides of powers we get 

By taking reciprocal on both sides we get,

Substituting in we get


By taking least common multiply we get 

Hence the correct choice is .

Question 11:

When simplified ${\left(-\frac{1}{27}\right)}^{-2/3}$is

(a) 9

(b) −9

(c) $\frac{1}{9}$

(d) $-\frac{1}{9}$

Answer 11:

We have to find the value of
So,


Hence the correct choice is .

Question 12:

Which one of the following is not equal to ${\left(\sqrt[3]{8}\right)}^{-1/2} ?$

(a) ${\sqrt[3]{2}}^{-1/2}$

(b) $8^{-1/6}$

(c) $\frac{1}{(\sqrt[3]{8}{)}^{1/2}}$

(d) $\frac{1}{\sqrt{2}}$

Answer 12:

We have to find the value of
So, 


Also,
Hence the correct alternative is .

Question 13:

Which one of the following is not equal to ${\left(\frac{100}{9}\right)}^{-3/2}$?

(a) ${\left(\frac{9}{100}\right)}^{3/2}$

(b) ${\left(\frac{1}{{\displaystyle \frac{100}{9}}}\right)}^{3/2}$

(c) $\frac{3}{10}\times \frac{3}{10}\times \frac{3}{10}$

(d) $\sqrt{\frac{100}{9}}\times \sqrt{\frac{100}{9}}\times \sqrt{\frac{100}{9}}$

Answer 13:

We have to find the value of
So,


Since, is equal to ,,.
Hence the correct choice is

Question 14:

If a, b, c are positive real numbers, then $\sqrt{a^{-1}b}\times \sqrt{b^{-1}c}\times \sqrt{c^{-1}a}$ is equal to

(a) 1

(b) abc

(c) $\sqrt{abc}$

(d) $\frac{1}{abc}$
 

Answer 14:

We have to find the value of when a, b, c are positive real numbers.
So,

Taking square root as common we get 
$$\begin{gathered} \sqrt{a^{-1}b}\times \sqrt{b^{-1}c}\times \sqrt{c^{-1}a}=\sqrt{\frac{b}{a}\times \frac{c}{b}\times \frac{a}{c}} \\ \sqrt{a^{-1}b}\times \sqrt{b^{-1}c}\times \sqrt{c^{-1}a}=1 \end{gathered}$$
Hence the correct alternative is .

Page-2.30

Question 15:

, then x =

(a) 2
(b) 3
(c) 4
(d) 1

Answer 15:

We have to find value of provided
So,


Equating exponents of power we get
Hence the correct alternative is

Question 16:

The value of ${\left\{8^{-4/3}÷2^{-2}\right\}}^{1/2}$ is

(a) $\frac{1}{2}$

(b) 2

(c) $\frac{1}{4}$

(d) 4

Answer 16:

Find the value of



Hence the correct choice is .

Question 17:

If a, b, c are positive real numbers, then $\sqrt[5]{3125a^{10}{b}^5{c}^{10}}$ is equal to

(a) 5a²bc²

(b) 25ab²c

(c) 5a³bc³

(d) 125a²bc²

Answer 17:

Find value of.

$\sqrt[5]{3125a^{10}{b}^5{c}^{10}}=5a^{2}b{c}^2$
Hence the correct choice is .

Question 18:

If a, m, n are positive ingegers, then ${\left\{\sqrt[m]{\sqrt[n]{a}}\right\}}^{mn}$is equal to

(a) a^mn

(b) a

(c) a<sup>m/n

(d) 1</sup>

Answer 18:

Find the value of .
So,



Hence the correct choice is

Question 19:

If x = 2 and y = 4, then ${\left(\frac{x}{y}\right)}^{x-y}+{\left(\frac{y}{x}\right)}^{y-x} =$

(a) 4

(b) 8

(c) 12

(d) 2

Answer 19:

We have to find the value of if,
Substitute,into get,


Hence the correct choice is .

Question 20:

The value of m for which ${\left[{\left\{{\left(\frac{1}{7^2}\right)}^{-2}\right\}}^{-1/3}\right]}^{1/4}=7^m,$ is

(a) $-\frac{1}{3}$

(b) $\frac{1}{4}$

(c) −3

(d) 2

Answer 20:

We have to find the value of for


By using rational exponents
$7^{\frac{-1}{3}}=7^m$
Equating power of exponents we get
Hence the correct choice is .

Question 21:

The value of ${\left\{{\left(23+2^2\right)}^{2/3}+(140-19{)}^{1/2}\right\}}^2,$ is

(a) 196

(b) 289

(c) 324

(d) 400

Answer 21:

We have to find the value of



By using the identity we get,

Hence correct choice is .

Question 22:

(256)^0.16 × (256)^0.09

(a) 4
(b) 16
(c) 64
(d) 256.25

Answer 22:

We have to find the value of. So,
By using law of rational exponents
we get


The value of is 4
Hence the correct choice is .

Question 23:

If 10^2y = 25, then 10^-y equals

(a) $-\frac{1}{5}$

(b) $\frac{1}{50}$

(c) $\frac{1}{625}$

(d) $\frac{1}{5}$

Answer 23:

We have to find the value of
Given that, therefore,


Hence the correct option is .

Question 24:

If 9^x+2 = 240 + 9^x, then x =

(a) 0.5
(b) 0.2
(c) 0.4
(d) 0.1

Answer 24:

We have to find the value of
Given



By equating the exponents we get 

Hence the correct alternative is .

Question 25:

If x is a positive real number and x² = 2, then x³ =

(a) $\sqrt{2}$

(b) 2$\sqrt{2}$

(c) 3$\sqrt{2}$

(d) 4

Answer 25:

We have to find provided. So,
By raising both sides to the power

By substituting in we get

The value of is
Hence the correct choice is .

Page-2.31

Question 26:

If $\frac{x}{x^{1.5}}=8x^{-1}$ and x > 0, then x =

(a) $\frac{\sqrt{2}}{4}$

(b) $\sqrt[2]{2}$

(c) 4

(d) 64 
 

Answer 26:

For, we have to find the value of x.
So,




By raising both sides to the power we get

The value of is
Hence the correct alternative is

Question 27:

If $g = t^{2/3}+4t^{-1/2}$, What is the value of g when t = 64?

(a) $\frac{21}{2}$

(b) $\frac{33}{2}$

(c) $16$

(d) $\frac{257}{16}$

Answer 27:

Given.We have to find the value of
So,



The value of is
Hence the correct choice is

Question 28:

If $4x - 4x^{-1} = 24,$then (2x)^x equals

(a) $5\sqrt{5}$

(b) $\sqrt{5}$

(c) $25\sqrt{5}$

(d) 125
 

Answer 28:

We have to find the value of if
So,
Taking as common factor we get 


By equating powers of exponents we get 

By substituting in we get


Hence the correct choice is

Question 29:

When simplified $\left(256\right){ }^{-\left(4^{-3/2}\right)}$ is

(a) 8

(b) $\frac{1}{8}$

(c) 2

(d) $\frac{1}{2}$

Answer 29:

Simplify

${\left(256\right)}^{{-}^{\left(4^{-\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}}={\left(256\right)}^{-{\left(2\right)}^{\left(-3\right)}}$



Hence the correct choice is .

Question 30:

If$\frac{3^{2x-8}}{225}=\frac{5^3}{5^x},$ then x =

(a) 2
(b) 3
(c) 5
(d) 4

Answer 30:

We have to find the value of provided
So,

By cross multiplication we get 

By equating exponents we get 

And 

Hence the correct choice is

Question 31:

The value of 64^-1/3 (64^1/3-64^2/3), is

(a) 1

(b) $\frac{1}{3}$

(c) −3

(d) −2

Answer 31:

Find the value of
So,


Hence the correct statement is.

Question 32:

If $\sqrt{5^n}=125$, then =

(a) 25

(b) $\frac{1}{125}$

(c) 625

(d) $\frac{1}{5}$

Answer 32:

We have to find provided
So,


Substitute in to get

Hence the value of is
The correct choice is

Question 33:

If (16)^2x+3 =(64)^x+3, then 4^2x-2 =

(a) 64

(b) 256

(c) 32

(d) 512

Answer 33:

We have to find the value ofprovided
So,

Equating the power of exponents we get


The value of is 


Hence the correct alternative is

Question 34:

If $2^{-m}\times \frac{1}{2^m}=\frac{1}{4},$ then $\frac{1}{14}\left\{(4^m{)}^{1/2}+{\left(\frac{1}{5^m}\right)}^{-1}\right\}$ is equal to

(a) $\frac{1}{2}$

(b) 2 

(c) 4

(d) $-\frac{1}{4}$

Answer 34:

We have to find the value ofprovided
Consider,

Equating the power of exponents we get 

By substituting we get 


Hence the correct choice is .

Question 35:

If $\frac{2^{m+n}}{2^{n-m}}=16$, $\frac{3^p}{3^n}=81$ and $a=2^{1/10}$, then$\frac{a^{2m+n-p}}{(a^{m-2n+2p}{)}^{-1}}=$

(a) 2

(b) $\frac{1}{4}$

(c) 9

(d) $\frac{1}{8}$

Answer 35:

Given :  , $\frac{3^p}{3^n}=81$ and  
To find :  
Find : 
By using rational components We get

By equating rational exponents we get 

Now,   =$\left(a^{2m+n-p}\right).\left(a^{m-2n+2p}\right)$  we get
$$\begin{gathered} ={\mathrm{a}}^{2\mathrm{m}+\mathrm{n}-\mathrm{p}+\mathrm{m}-2\mathrm{n}+2\mathrm{p}} \\ ={\mathrm{a}}^{3\mathrm{m}-\mathrm{n}+\mathrm{p}} \\ \mathrm{Now} \mathrm{putting} \mathrm{value} \mathrm{of} \mathrm{a} = 2^{\frac{1}{10}} \mathrm{we} \mathrm{get}, \\ =2^{\frac{3\mathrm{m}-\mathrm{n}+\mathrm{p}}{10}} \\ =2^{\frac{6-\mathrm{n}+\mathrm{p}}{10}} \end{gathered}$$

Also, $\frac{3^p}{3^n}=81$
$3^{p-n}=3^4$
On comparing LHS and RHS we get, p - n = 4.
Now,
= a^{3m - n + p
$$\begin{gathered} =2^{\frac{6+(p-n)}{10}} \\ =2^{\frac{6+4}{10}} \\ =2^{\frac{10}{10}}=2^1 \\ =2 \end{gathered}$$}

So, option (a) is the correct answer.

Page-2.32

Question 36:

If $\frac{3^{5x}\times {81}^2\times 6561}{3^{2x}}=3^7$, then x =

(a) 3

(b) −3

(c) $\frac{1}{3}$

(d) $-\frac{1}{3}$

Answer 36:

We have to find the value of x provided
So,

By using law of rational exponents we get

By equating exponents we get


Hence the correct choice is .

Question 37:

If o <y <x, which statement must be true?

(a) $\sqrt{x}-\sqrt{y}=\sqrt{x-y}$

(b) $\sqrt{x} + \sqrt{x} = \sqrt{2x}$

(c) $x\sqrt{y}=y\sqrt{x}$

(d) $\sqrt{xy} =\sqrt{x}\sqrt{y}$

Answer 37:

We have to find which statement must be true?
Given
Option (a) :
Left hand side:

Right Hand side:

Left hand side is not equal to right hand side 
The statement is wrong. 
Option (b) : 

Left hand side:

Right Hand side:

Left hand side is not equal to right hand side 
The statement is wrong.
Option (c) : 

Left hand side:

Right Hand side:

Left hand side is not equal to right hand side 
The statement is wrong. 
Option (d) : 

Left hand side: 

Right Hand side:

Left hand side is equal to right hand side 
The statement is true.
Hence the correct choice is .

Question 38:

If 10^x = 64, what is the value of ${10}^{\frac{x}{2}+1} ?$

(a) 18
(b) 42
(c) 80
(d) 81

Answer 38:

We have to find the value of provided
So,

By substituting we get 

Hence the correct choice is .

Question 39:

$\frac{5^{n+2}-6\times 5^{n+1}}{13\times 5^n-2\times 5^{n+1}}$ is equal to

(a) $\frac{5}{3}$

(b) $-\frac{5}{3}$

(c) $\frac{3}{5}$

(d) $-\frac{3}{5}$


 

Answer 39:

We have to simplify
Taking as a common factor we get

Hence the correct alternative is

Question 40:

If $\sqrt{2^n}=1024,$ then ${3^2}^{\left(\frac{n}{4}-4\right)}=$

(a) 3

(b) 9

(c) 27

(d) 81

Answer 40:

We have to find
Given

Equating powers of rational exponents we get 

Substituting in we get 

Hence the correct choice is .