Rd Sharma Solutions for Class 9 Math Chapter 2 Exponents Of Real Numbers are provided here with simple step-by-step explanations. These solutions for Exponents Of Real Numbers are extremely popular among Class 9 students for Math Exponents Of Real Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Book of Class 9 Math Chapter 2 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Solutions. All Rd Sharma Solutions for class Class 9 Math are prepared by experts and are 100% accurate.

Page No 2.12:

Question 1:

Assuming that x, y, z are positive real numbers, simplify each of the following:

(i) x-35

(ii) x3 y-2

(iii) (x-2/3y-1/2)2

(iv) (x)-2/3y4÷xy-1/2

(v) 243 x10y5z105

(vi) x-4y-105/4

(vii) 235672

Answer:

We have to simplify the following, assuming thatare positive real numbers

(i) Given

As x is positive real number then we have

Hence the simplified value of is

(ii) Given

As x and y are positive real numbers then we can write 

By using law of rational exponents we have 

Hence the simplified value of is

(iii) Given

As x and y are positive real numbers then we have 

By using law of rational exponents we have 

By using law of rational exponents we have


x-23y-122=1x23+23×1y12+12=1x43×1y22=1x43×1y=1x43y

Hence the simplified value of is .


 iv x-23 y4 ÷ xy-12=x12-23 y412 ÷ x × y-1212=x12×-23 × y4×12x12 × y-12×12    =x-13 × y2x12 × y-14by using the law of rational exponents, am ÷ an = am-n, we have     x-13-12 × y2+14=x-56 × y94            =y94x56v. 243  x10 y5 z105=243 × x10 × y5 × z1015=24315 × x1015 × y515 × z1015=3515 × x10×15 × y5×15 × z10×15=3 × x2 × y × z2=3x2yz2vi x-4y-1054=x-454y-1054=x-4×54y-10×54=x-5y-252=y252x5

(vii) 235672

235672=232+2+1672=232×232×231×672=23×23×231×672
=23×23×231×672=162493=5127203=512720312

 

Page No 2.12:

Question 2:

Simplify:

(i) (16-1/5)5/2

(ii) 32-35

(iii) (343)-23

(iv) (0.001)1/3

(v) (25)3/2×(243)3/5(16)5/4×(8)4/3

(vi) 258 ÷ 2513

(vii) 5-1×7252×7-47/2 × 5-2×7353×7-5-5/2

Answer:

(1) Given

By using law of rational exponents we have

Hence the value of is
(ii) 
32-35

=13235=13235=12535=125×35
=123=18

(iii) Given

Hence the value of is

(iv) Given

The value of is

(v) Given

Hence the value of is

(vi) Given. So,

By using the law of rational exponents

Hence the value of is

(vii) Given . So,

Hence the value of is

Page No 2.12:

Question 3:

Prove that:

(i) 3×5-3 ÷ 3-13 5×3×566=35

(ii) 93/2-3×50-181-1/2 = 15

(iii) 14-2-3×82/3×40+916-1/2 = 163

(iv) 21/2×31/3×41/410-1/5×53/5 ÷ 34/3×5-7/54-3/5×6=10

(v) 14 +(0.01)-1/2-(27)2/3 =32

(vi) 2n + 2n -12n+1-2n=32

(vii) 64125-2/3 + 12566251/4 + 25643 = 6516

(viii) 3-3×62×9852×1/253×(15)-4/3×31/3=282

(ix) (0.6)0-(0.1)-138-1323+-13-1=-32

Answer:

(i) We have to prove that

By using rational exponent we get,

Hence,

(ii) We have to prove that. So,

Hence,

(iii) We have to prove that

Now,

Hence,

(iv) We have to prove that. So,

Let

Hence,

(v) We have to prove that

Let

 

Hence,

(vi) We have to prove that . So,

Let

Hence,

(vii) We have to prove that. So let

By taking least common factor we get 

 

Hence,

(viii) We have to prove that. So,

Let

Hence,

(ix) We have to prove that. So,

Let

Hence,

Page No 2.12:

Question 4:

If 27x =93x,  find x.

Answer:

We are given. We have to find the value of

Since

By using the law of exponents we get, 

On equating the exponents we get,

Hence,

Page No 2.12:

Question 5:

Find the values of x in each of the following:

(i) 25x÷2x =2205

(ii) (23)4=(22)x

(iii) 35x 532x=12527

(iv) 5x-2×32x-3 =135

(v) 2x-7×5x-4=1250

(vi) 432x+12=132

(vii) 52x+3=1

(viii) 13x=44-34-6

(ix) 35x+1=12527

Answer:

From the following we have to find the value of x

(i) Given

By using rational exponents

On equating the exponents we get,

The value of x is

(ii) Given

On equating the exponents 

Hence the value of x is

(iii) Given

Comparing exponents we have,

Hence the value of x is

(iv) Given

On equating the exponents of 5 and 3 we get,

And,

The value of x is

(v) Given

On equating the exponents we get 

And, 

Hence the value of x is

(vi) 
432x+12=132

22134x+12=12524x+13=2-5

On comparing we get, 
4x+13=-54x+1=-154x=-16x=-4

(vii) 52x+3=1

52x+3=502x+3=0x=-32

(viii) 13x=44-34-6

13x=224-34-613x=28-34-613x=256-81-613x=16913x=132

On comparing we get, 
x=2on squaring both sides we get, x=4

(ix) 35x+1=12527
35x+1=53335x+12=35-3

On comparing we get, 

x+12=-3x+1=-6x=-7



Page No 2.13:

Question 1:

Write 625-1/4 in decimal form.

Answer:

We have to writein decimal form. So,

625-14=162514=15414

Hence the decimal form of is

Page No 2.13:

Question 2:

State the product law of exponents.

Answer:

State the product law of exponents.

If is any real number and , are positive integers, then

By definition, we have 

(factor) ( factor)

to factors

Thus the exponent "product rule" tells us that, when multiplying two powers that have the same base, we can add the exponents. 

Page No 2.13:

Question 3:

State the quotient law of exponents.

Answer:

The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If a is a non-zero real number and m, n are positive integers, then

We shall divide the proof into three parts 

(i) when

(ii) when

(iii) when

Case 1 

When

We have 

aman=a×a×a....to m factorsa×a×a....to n factorsaman=a×a×a....to (m-n) factorsaman=am-n

Case 2 

When

We get

Cancelling common factors in numerator and denominator we get,

By definition we can write 1 as

Case 3 

When

In this case, we have 

Hence, whether, or,

Page No 2.13:

Question 4:

State the power law of exponents.

Answer:

The "power rule" tell us that to raise a power to a power, just multiply the exponents. 

If a is any real number and m, n are positive integers, then

We have,

factors

factors

Hence,

Page No 2.13:

Question 5:

For any positive real number x, find the value of

xaxba+b×xbxcb+c×xcxac+a

Answer:

We have to find the value of L =

By using rational exponents, we get

By using rational exponents we get 

By definition we can write as 1

Hence the value of expression is .

Page No 2.13:

Question 6:

Write the value of 5(81/3+271/3)31/4.

Answer:

We have to find the value of. So,

By using rational exponents we get 

Hence the simplified value of is

Page No 2.13:

Question 7:

Simplify 625-1/2-1/42

Answer:

We have to simplify. So,

Hence, the value of is

Page No 2.13:

Question 8:

For any positive real number x, write the value of

xab1abxbc1bcxca1ca

Answer:

We have to simplify. So,

By using rational exponents, we get

Hence the value of is



Page No 2.14:

Question 9:

If (− 1)3 = 8, What is the value of (+ 1)2 ?

Answer:

We have to find the value of , where 

Consider

By equating the base, we get

By substituting in

Hence the value of is .

Page No 2.14:

Question 10:

If 24 × 42 =16x, then find the value of x.

Answer:

We have to find the value of x provided

So,

By equating the exponents we get

Hence the value of x is .

Page No 2.14:

Question 11:

If 3x-1 = 9 and 4y+2 = 64, what is the value of xy ?

Answer:

We have to find the value of for

So,

By equating the exponent we get

Let’s take

By equating the exponent we get

By substituting in we get

Hence the value of is

Page No 2.14:

Question 12:

Write the value of 73×493.

Answer:

We have to find the value of . So,

By using law rational exponents we get,

Hence the value of is

Page No 2.14:

Question 13:

Write 19-1/2×(64)-1/3 as a rational number.

Answer:

We have to find the value of . So,

Hence the value of the value of is .

Page No 2.14:

Question 14:

Write the value of 125×273.

Answer:

We have to find the value of 125×273. So,


125×273=53×333=5×3=15

Hence the value of the value of is .

Page No 2.14:

Question 1:

The value of 2-3 (2-3)33 is

(a) 5

(b) 125

(c) 1/5

(d) -125

Answer:

We have to find the value of. So,

The value of is 125

Hence the correct choice is

Page No 2.14:

Question 2:

(256)0.16 × (256)0.09

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

Answer:

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 .

Page No 2.14:

Question 3:

If 102y = 25, then 10-y equals

(a) -15

(b) 150

(c) 1625

(d) 15

Answer:

We have to find the value of

Given that, therefore,

Hence the correct option is .

Page No 2.14:

Question 4:

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

(a) 18
(b) −18
(c) 14
(d) −14

Answer:

Given

Here

By substituting in we get 

The value of is – 14

Hence the correct choice is .

Page No 2.14:

Question 5:

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:

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

=x3+26=x56

The product of the square root of x with the cube root of x is

Hence the correct alternative is

Page No 2.14:

Question 6:

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

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

Answer:

We have to find the value of

Given

By equating the exponents we get 

Hence the correct alternative is .

Page No 2.14:

Question 7:

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

(b) x

(c) x56

(d) 1x56

Answer:

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 .

Page No 2.14:

Question 8:

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

(a) 64
(b) 2
(c) 12
(d) 642/3

Answer:

We have to find the value of

So,

The value of is

Hence the correct choice is .

Page No 2.14:

Question 9:

Which of the following is (are) not equal to 561/5-1/6 ?
(a) 5615-36 

(b) 1561/51/6

(c) 651/30

(d) 56-1/30

Answer:

We have to find the value of

So,

Hence the correct choice is .



Page No 2.15:

Question 10:

When simplified (x-1+y-1)-1 is equal to

(a) xy

(b) x+y

(c) xyx+y

(d) x+yxy

Answer:

We have to simplify

So,

The value of is

Hence the correct choice is .

Page No 2.15:

Question 11:

If 8x+1 = 64 , what is the value of 32x+1 ?

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

Answer:

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 .

Page No 2.15:

Question 12:

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

(a) x-y=x-y

(b) x + x = 2x

(c) xy=yx

(d) xy =xy

Answer:

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 .

Page No 2.15:

Question 13:

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

(a) 2

(b) 22

(c) 32

(d) 4

Answer:

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 No 2.15:

Question 14:

If (23)2 = 4x, then 3x =

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

Answer:

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

Page No 2.15:

Question 15:

If 10x = 64, what is the value of 10x2+1 ?

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

Answer:

We have to find the value of provided

So,

By substituting we get 

Hence the correct choice is .

Page No 2.15:

Question 16:

If xx1.5=8x-1 and x > 0, then x =

(a) 24

(b) 22

(c) 4

(d) 64  
 

Answer:

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

Page No 2.15:

Question 17:

If g = t2/3+4t-1/2, What is the value of g when t = 64?

(a) 212

(b) 332

(c) 16

(d) 25716

Answer:

Given.We have to find the value of

So,

The value of is

Hence the correct choice is

Page No 2.15:

Question 18:

If x-2 = 64, then x1/3+x0 =

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

Answer:

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 .

Page No 2.15:

Question 19:

If 4x - 4x-1 = 24, then (2x)x equals

(a) 55

(b) 5

(c) 255

(d) 125
 

Answer:

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

Page No 2.15:

Question 20:

When simplified -127-2/3 is

(a) 9

(b) −9

(c) 19

(d) -19

Answer:

We have to find the value of

So,

Hence the correct choice is .



Page No 2.16:

Question 21:

Which one of the following is not equal to 83-1/2 ?

(a) 23-1/2

(b) 8-1/6

(c) 1(83)1/2

(d) 12

Answer:

We have to find the value of

So, 

Also,

Hence the correct alternative is .

Page No 2.16:

Question 22:

Which one of the following is not equal to 1009-3/2 ?

(a) 91003/2

(b) 110093/2

(c) 310×310×310

(d) 1009×1009×1009

Answer:

We have to find the value of

So,

Since, is equal to ,,.

Hence the correct choice is

Page No 2.16:

Question 23:

When simplified (256) -(4-3/2) is  

(a) 8

(b) 18

(c) 2

(d) 12

Answer:

Simplify


256-4-32=256-2-3
 

Hence the correct choice is .

Page No 2.16:

Question 24:

5n+2-6×5n+113×5n-2×5n+1 is equal to

(a) 53

(b) -53

(c) 35

(d) -35


 

Answer:

We have to simplify

Taking as a common factor we get

Hence the correct alternative is

Page No 2.16:

Question 25:

If a, b, c are positive real numbers, then a-1b×b-1c×c-1a is equal to

(a) 1

(b) abc

(c) abc

(d) 1abc
 

Answer:

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

So,

Taking square root as common we get 

a-1b×b-1c×c-1a=ba×cb×aca-1b×b-1c×c-1a=1

Hence the correct alternative is .

Page No 2.16:

Question 26:

If32x-8225=535x, then x =

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

Answer:

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

Page No 2.16:

Question 27:

, then x =

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

Answer:

We have to find value of provided

So,

Equating exponents of power we get

Hence the correct alternative is

Page No 2.16:

Question 28:

The value of 8-4/3÷2-21/2 is

(a) 12

(b) 2

(c) 14

(d) 4

Answer:

Find the value of

Hence the correct choice is .

Page No 2.16:

Question 29:

If a, b, c are positive real numbers, then 3125a10b5c105 is equal to

(a) 5a2bc2

(b) 25ab2c

(c) 5a3bc3

(d) 125a2bc2

Answer:

Find value of.

3125a10b5c105=5a2bc2

Hence the correct choice is .

Page No 2.16:

Question 30:

The value of 64-1/3 (641/3-642/3), is

(a) 1

(b) 13

(c) −3

(d) −2

Answer:

Find the value of

So,

Hence the correct statement is.



Page No 2.17:

Question 31:

If 5n=125, then =

(a) 25

(b) 1125

(c) 625

(d) 15

Answer:

We have to find provided

So,

Substitute in to get

Hence the value of is

The correct choice is

Page No 2.17:

Question 32:

If (16)2x+3 =(64)x+3, then 42x-2 =

(a) 64

(b) 256

(c) 32

(d) 512

Answer:

We have to find the value ofprovided

So,

Equating the power of exponents we get

The value of is 

Hence the correct alternative is

Page No 2.17:

Question 33:

If a, m, n are positive ingegers, then anmmnis equal to

(a) amn

(b) a

(c) am/n

(d) 1

Answer:

Find the value of .

So,

Hence the correct choice is

Page No 2.17:

Question 34:

If 2-m×12m=14, then 114(4m)1/2+15m-1 is equal to

(a) 12

(b) 2  

(c) 4

(d) -14

Answer:

We have to find the value ofprovided

Consider,

Equating the power of exponents we get 

By substituting we get 

Hence the correct choice is .

Page No 2.17:

Question 35:

If x = 2 and y = 4, then xyx-y+yxy-x =

(a) 4

(b) 8

(c) 12

(d) 2

Answer:

We have to find the value of if,

Substitute,into get,

Hence the correct choice is .

Page No 2.17:

Question 36:

The value of m for which 172-2-1/31/4=7m, is

(a) -13

(b) 14

(c) −3

(d) 2

Answer:

We have to find the value of for

By using rational exponents

7-13=7m

Equating power of exponents we get

Hence the correct choice is .

Page No 2.17:

Question 37:

If 2m+n2n-m=163p3n=81 and a=21/10, thena2m+n-p(am-2n+2p)-1=

(a) 2

(b) 14

(c) 9

(d) 18

Answer:

Given :  3p3n=81 and  
To find :  

Find : 
By using rational components We get

By equating rational exponents we get 

Now,   =a2m+n-p.am-2n+2p  we get
=a2m+n-p+m-2n+2p=a3m-n+pNow putting value of a = 2110 we get, =23m-n+p10=26-n+p10

Also, 3p3n=81
3p-n=34
On comparing LHS and RHS we get, p - n = 4.
Now, 
= a3m - n + p
=26+(p-n)10=26+410=21010=21=2


So, option (a) is the correct answer.

 

Page No 2.17:

Question 38:

The value of 23+222/3+(140-19)1/22,  is

(a) 196

(b) 289

(c) 324

(d) 400

Answer:

We have to find the value of

By using the identity we get,

Hence correct choice is .

Page No 2.17:

Question 39:

If 2n=1024,  then 32n4-4=

(a) 3

(b) 9

(c) 27

(d) 81

Answer:

We have to find

Given

Equating powers of rational exponents we get 

Substituting in we get 

Hence the correct choice is .

Page No 2.17:

Question 40:

If 35x×812×656132x=37, then x =

(a) 3

(b) −3

(c) 13

(d) -13

Answer:

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 .



View NCERT Solutions for all chapters of Class 9