Chapters
Chapter 2: Exponents of Real Numbers
Chapter 3: Rationalisation
Chapter 4: Algebraic Identities
Chapter 5: Factorisation of Algebraic Expressions
Chapter 6: Factorisation of Polynomials
Chapter 7: Linear Equations in Two Variables
Chapter 8: Co-ordinate Geometry
Chapter 9: Introduction to Euclid’s Geometry
Chapter 10: Lines and Angles
Chapter 11: Triangle and its Angles
Chapter 12: Congruent Triangles
Chapter 13: Quadrilaterals
Chapter 14: Areas of Parallelograms and Triangles
Chapter 15: Circles
Chapter 16: Constructions
Chapter 17: Heron’s Formula
Chapter 18: Surface Areas and Volume of a Cuboid and Cube
Chapter 19: Surface Areas and Volume of a Circular Cylinder
Chapter 20: Surface Areas and Volume of A Right Circular Cone
Chapter 21: Surface Areas and Volume of a Sphere
Chapter 22: Tabular Representation of Statistical Data
Chapter 23: Graphical Representation of Statistical Data
Chapter 24: Measures of Central Tendency
Chapter 25: Probability

Chapter 2: Exponents of Real Numbers
RD Sharma solutions for Mathematics for Class 9 Chapter 2 Exponents of Real Numbers Exercise 2.1 [Pages 12 - 13]
Simplify the following
`3(a^4b^3)^10xx5(a^2b^2)^3`
Simplify the following
`(2x^-2y^3)^3`
Simplify the following
`((4xx10^7)(6xx10^-5))/(8xx10^4)`
Simplify the following
`(4ab^2(-5ab^3))/(10a^2b^2)`
Simplify the following
`((x^2y^2)/(a^2b^3))^n`
Simplify the following
`(a^(3n-9))^6/(a^(2n-4))`
If a = 3 and b = -2, find the values of :
aa + bb
If a = 3 and b = -2, find the values of :
ab + ba
If a = 3 and b = -2, find the values of :
(a + b)ab
Prove that:
`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`
Prove that:
`(x^a/x^b)^cxx(x^b/x^c)^axx(x^c/x^a)^b=1`
Prove that:
`1/(1+x^(a-b))+1/(1+x^(b-a))=1`
Prove that:
`1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))=1`
Prove that:
`(a+b+c)/(a^-1b^-1+b^-1c^-1+c^-1a^-1)=abc`
Prove that:
`(a^-1+b^-1)^-1=(ab)/(a+b)`
If abc = 1, show that `1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)=1`
Simplify the following:
`(3^nxx9^(n+1))/(3^(n-1)xx9^(n-1))`
Simplify the following:
`(5xx25^(n+1)-25xx5^(2n))/(5xx5^(2n+3)-25^(n+1))`
Simplify the following:
`(5^(n+3)-6xx5^(n+1))/(9xx5^x-2^2xx5^n)`
Simplify the following:
`(6(8)^(n+1)+16(2)^(3n-2))/(10(2)^(3n+1)-7(8)^n)`
Solve the following equation for x:
`7^(2x+3)=1`
Solve the following equation for x:
`2^(x+1)=4^(x-3)`
Solve the following equation for x:
`2^(5x+3)=8^(x+3)`
Solve the following equation for x:
`4^(2x)=1/32`
Solve the following equation for x:
`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`
Solve the following equation for x:
`2^(3x-7)=256`
Solve the following equations for x:
`2^(2x)-2^(x+3)+2^4=0`
Solve the following equations for x:
`3^(2x+4)+1=2.3^(x+2)`
If 49392 = a4b2c3, find the values of a, b and c, where a, b and c are different positive primes.
If `1176=2^a3^b7^c,` find a, b and c.
Given `4725=3^a5^b7^c,` find
(i) the integral values of a, b and c
(ii) the value of `2^-a3^b7^c`
If `a=xy^(p-1), b=xy^(q-1)` and `c=xy^(r-1),` prove that `a^(q-r)b^(r-p)c^(p-q)=1`
RD Sharma solutions for Mathematics for Class 9 Chapter 2 Exponents of Real Numbers Exercise 2.2 [Pages 24 - 27]
Assuming that x, y, z are positive real numbers, simplify the following:
`(sqrt(x^-3))^5`
Assuming that x, y, z are positive real numbers, simplify the following:
`sqrt(x^3y^-2)`
Assuming that x, y, z are positive real numbers, simplify the following:
`(x^((-2)/3)y^((-1)/2))^2`
Assuming that x, y, z are positive real numbers, simplify the following:
`(sqrtx)^((-2)/3)sqrt(y^4)divsqrt(xy^((-1)/2))`
Assuming that x, y, z are positive real numbers, simplify the following:
`root5(243x^10y^5z^10)`
Assuming that x, y, z are positive real numbers, simplify the following:
`(x^-4/y^-10)^(5/4)`
Assuming that x, y, z are positive real numbers, simplify the following:
`(sqrt2/sqrt3)^5(6/7)^2`
Simplify:
`(16^(-1/5))^(5/2)`
Simplify:
`root5((32)^-3)`
Simplify:
`root3((343)^-2)`
Simplify:
`(0.001)^(1/3)`
Simplify:
`((25)^(3/2)xx(243)^(3/5))/((16)^(5/4)xx(8)^(4/3))`
Simplify:
`(sqrt2/5)^8div(sqrt2/5)^13`
Simplify:
`((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)`
Prove that:
`sqrt(3xx5^-3)divroot3(3^-1)sqrt5xxroot6(3xx5^6)=3/5`
Prove that:
`9^(3/2)-3xx5^0-(1/81)^(-1/2)=15`
Prove that:
`(1/4)^-2-3xx8^(2/3)xx4^0+(9/16)^(-1/2)=16/3`
Prove that:
`(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10`
Prove that:
`sqrt(1/4)+(0.01)^(-1/2)-(27)^(2/3)=3/2`
Prove that:
`(2^n+2^(n-1))/(2^(n+1)-2^n)=3/2`
Prove that:
`(64/125)^(-2/3)+1/(256/625)^(1/4)+(sqrt25/root3 64)=65/16`
Prove that:
`(3^-3xx6^2xxsqrt98)/(5^2xxroot3(1/25)xx(15)^(-4/3)xx3^(1/3))=28sqrt2`
Prove that:
`((0.6)^0-(0.1)^-1)/((3/8)^-1(3/2)^3+((-1)/3)^-1)=(-3)/2`
Show that:
`1/(1+x^(a-b))+1/(1+x^(b-a))=1`
Show that:
`[{x^(a(a-b))/x^(a(a+b))}div{x^(b(b-a))/x^(b(b+a))}]^(a+b)=1`
Show that:
`(x^(1/(a-b)))^(1/(a-c))(x^(1/(b-c)))^(1/(b-a))(x^(1/(c-a)))^(1/(c-b))=1`
Show that:
`(x^(a^2+b^2)/x^(ab))^(a+b)(x^(b^2+c^2)/x^(bc))^(b+c)(x^(c^2+a^2)/x^(ac))^(a+c)=x^(2(a^3+b^3+c^3))`
Show that:
`(x^(a-b))^(a+b)(x^(b-c))^(b+c)(x^(c-a))^(c+a)=1`
Show that:
`{(x^(a-a^-1))^(1/(a-1))}^(a/(a+1))=x`
Show that:
`(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)=1`
Show that:
`(3^a/3^b)^(a+b)(3^b/3^c)^(b+c)(3^c/3^a)^(c+a)=1`
If 2x = 3y = 12z, show that `1/z=1/y+2/x`
If 2x = 3y = 6-z, show that `1/x+1/y+1/z=0`
If ax = by = cz and b2 = ac, show that `y=(2zx)/(z+x)`
If 3x = 5y = (75)z, show that `z=(xy)/(2x+y)`
If `27^x=9/3^x,` find x.
Find the value of x in the following:
`2^(5x)div2x=root5(2^20)`
Find the value of x in the following:
`(2^3)^4=(2^2)^x`
Find the value of x in the following:
`(3/5)^x(5/3)^(2x)=125/27`
Find the value of x in the following:
`5^(x-2)xx3^(2x-3)=135`
Find the value of x in the following:
`2^(x-7)xx5^(x-4)=1250`
Find the value of x in the following:
`(root3 4)^(2x+1/2)=1/32`
Find the value of x in the following:
`5^(2x+3)=1`
Find the value of x in the following:
`(13)^(sqrtx)=4^4-3^4-6`
Find the value of x in the following:
`(sqrt(3/5))^(x+1)=125/27`
If `x=2^(1/3)+2^(2/3),` Show that x3 - 6x = 6
Determine `(8x)^x,`If `9^(x+2)=240+9^x`
If `3^(x+1)=9^(x-2),` find the value of `2^(1+x)`
If `3^(4x) = (81)^-1` and `10^(1/y)=0.0001,` find the value of ` 2^(-x+4y)`.
If `5^(3x)=125` and `10^y=0.001,` find x and y.
Solve the following equation:
`3^(x+1)=27xx3^4`
Solve the following equation:
`4^(2x)=(root3 16)^(-6/y)=(sqrt8)^2`
Solve the following equation:
`3^(x-1)xx5^(2y-3)=225`
Solve the following equation:
`8^(x+1)=16^(y+2)` and, `(1/2)^(3+x)=(1/4)^(3y)`
Solve the following equation:
`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`
Solve the following equation:
`sqrt(a/b)=(b/a)^(1-2x),` where a and b are distinct primes.
If a and b are distinct primes such that `root3 (a^6b^-4)=a^xb^(2y),` find x and y.
If a and b are different positive primes such that
`((a^-1b^2)/(a^2b^-4))^7div((a^3b^-5)/(a^-2b^3))=a^xb^y,` find x and y.
If a and b are different positive primes such that
`(a+b)^-1(a^-1+b^-1)=a^xb^y,` find x + y + 2.
If `2^x xx3^yxx5^z=2160,` find x, y and z. Hence, compute the value of `3^x xx2^-yxx5^-z.`
If 1176 = `2^axx3^bxx7^c,` find the values of a, b and c. Hence, compute the value of `2^axx3^bxx7^-c` as a fraction.
Simplify:
`(x^(a+b)/x^c)^(a-b)(x^(b+c)/x^a)^(b-c)(x^(c+a)/x^b)^(c-a)`
Simplify:
`root(lm)(x^l/x^m)xxroot(mn)(x^m/x^n)xxroot(nl)(x^n/x^l)`
Show that:
`((a+1/b)^mxx(a-1/b)^n)/((b+1/a)^mxx(b-1/a)^n)=(a/b)^(m+n)`
If `a=x^(m+n)y^l, b=x^(n+l)y^m` and `c=x^(l+m)y^n,` Prove that `a^(m-n)b^(n-l)c^(l-m)=1`
If `x = a^(m+n),` `y=a^(n+l)` and `z=a^(l+m),` prove that `x^my^nz^l=x^ny^lz^m`
RD Sharma solutions for Mathematics for Class 9 Chapter 2 Exponents of Real Numbers Exercise 2.3 [Pages 28 - 29]
Write \[\left( 625 \right)^{- 1/4}\] in decimal form.
State the product law of exponents.
State the quotient law of exponents.
State the power law of exponents.
If 24 × 42 =16x, then find the value of x.
If 3x-1 = 9 and 4y+2 = 64, what is the value of \[\frac{x}{y}\] ?
Write the value of \[\sqrt[3]{7} \times \sqrt[3]{49} .\]
Write \[\left( \frac{1}{9} \right)^{- 1/2} \times (64 )^{- 1/3}\] as a rational number.
Write the value of \[\sqrt[3]{125 \times 27}\].
For any positive real number x, find the value of \[\left( \frac{x^a}{x^b} \right)^{a + b} \times \left( \frac{x^b}{x^c} \right)^{b + c} \times \left( \frac{x^c}{x^a} \right)^{c + a}\].
Write the value of \[\left\{ 5( 8^{1/3} + {27}^{1/3} )^3 \right\}^{1/4} . \]
Simplify \[\left[ \left\{ \left( 625 \right)^{- 1/2} \right\}^{- 1/4} \right]^2\]
For any positive real number x, write the value of \[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\]
If (x − 1)3 = 8, What is the value of (x + 1)2 ?
RD Sharma solutions for Mathematics for Class 9 Chapter 2 Exponents of Real Numbers Exercise 2.4 [Pages 29 - 33]
The value of \[\left\{ 2 - 3 (2 - 3 )^3 \right\}^3\] is
5
125
1/5
-125
The value of x − yx-y when x = 2 and y = −2 is
18
-18
14
-14
The product of the square root of x with the cube root of x is
cube root of the square root of x
sixth root of the fifth power of x
fifth root of the sixth power of x
sixth root of x
The seventh root of x divided by the eighth root of x is
x
\[\sqrt{x}\]
\[\sqrt[56]{x}\]
\[\frac{1}{\sqrt[56]{x}}\]
The square root of 64 divided by the cube root of 64 is
64
2
\[\frac{1}{2}\]
642/3
Which of the following is (are) not equal to \[\left\{ \left( \frac{5}{6} \right)^{1/5} \right\}^{- 1/6}\] ?
\[\left\{ \left( \frac{5}{6} \right)^\frac{1}{5} \right\}^{- \frac{3}{6}}\]
\[\frac{1}{\left\{ \left( \frac{5}{6} \right)^{1/5} \right\}^{1/6}}\]
\[\left( \frac{6}{5} \right)^{1/30}\]
\[\left( \frac{5}{6} \right)^{- 1/30}\]
When simplified \[( x^{- 1} + y^{- 1} )^{- 1}\] is equal to
xy
x+y
\[\frac{xy}{y + x}\]
\[\frac{x + y}{xy}\]
If \[8^{x + 1}\] = 64 , what is the value of \[3^{2x + 1}\] ?
1
3
9
27
If (23)2 = 4x, then 3x =
3
6
9
27
If x-2 = 64, then x1/3+x0 =
2
3
3/2
2/3
When simplified \[\left( - \frac{1}{27} \right)^{- 2/3}\] is
9
-9
\[\frac{1}{9}\]
\[- \frac{1}{9}\]
Which one of the following is not equal to \[\left( \sqrt[3]{8} \right)^{- 1/2} ?\]
\[\sqrt[3]{2}^{- 1/2}\]
\[8^{- 1/6}\]
\[\frac{1}{(\sqrt[3]{8} )^{1/2}}\]
\[\frac{1}{\sqrt{2}}\]
Which one of the following is not equal to \[\left( \frac{100}{9} \right)^{- 3/2}\]?
\[\left( \frac{9}{100} \right)^{3/2}\]
\[\left( \frac{1}{\frac{100}{9}} \right)^{3/2}\]
\[\frac{3}{10} \times \frac{3}{10} \times \frac{3}{10}\]
\[\sqrt{\frac{100}{9}} \times \sqrt{\frac{100}{9}} \times \sqrt{\frac{100}{9}}\]
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
1
abc
\[\sqrt{abc}\]
\[\frac{1}{abc}\]
`(2/3)^x (3/2)^(2x)=81/16 `then x =
2
3
4
1
The value of \[\left\{ 8^{- 4/3} \div 2^{- 2} \right\}^{1/2}\] is
\[\frac{1}{2}\]
2
\[\frac{1}{4}\]
4
If a, b, c are positive real numbers, then \[\sqrt[5]{3125 a^{10} b^5 c^{10}}\] is equal to
5a2bc2
25ab2c
5a3bc3
125a2bc2
If a, m, n are positive ingegers, then \[\left\{ \sqrt[m]{\sqrt[n]{a}} \right\}^{mn}\] is equal to
amn
a
am/n
1
If x = 2 and y = 4, then \[\left( \frac{x}{y} \right)^{x - y} + \left( \frac{y}{x} \right)^{y - x} =\]
4
8
12
2
The value of m for which \[\left[ \left\{ \left( \frac{1}{7^2} \right)^{- 2} \right\}^{- 1/3} \right]^{1/4} = 7^m ,\] is
\[- \frac{1}{3}\]
\[\frac{1}{4}\]
-3
2
The value of \[\left\{ \left( 23 + 2^2 \right)^{2/3} + (140 - 19 )^{1/2} \right\}^2 ,\] is
196
289
324
400
(256)0.16 × (256)0.09
4
16
64
256.25
If 102y = 25, then 10-y equals
- \[- \frac{1}{5}\]
- \[\frac{1}{50}\]
- \[\frac{1}{625}\]
- \[\frac{1}{5}\]
If 9x+2 = 240 + 9x, then x =
0.5
0.2
0.4
0.1
If x is a positive real number and x2 = 2, then x3 =
\[\sqrt{2}\]
2\[\sqrt{2}\]
3\[\sqrt{2}\]
4
If \[\frac{x}{x^{1 . 5}} = 8 x^{- 1}\] and x > 0, then x =
\[\frac{\sqrt{2}}{4}\]
\[\sqrt[2]{2}\]
4
64
If \[g = t^{2/3} + 4 t^{- 1/2}\] What is the value of g when t = 64?
\[\frac{21}{2}\]
\[\frac{33}{2}\]
\[16\]
\[\frac{257}{16}\]
If \[4x - 4 x^{- 1} = 24,\] then (2x)x equals
\[5\sqrt{5}\]
\[\sqrt{5}\]
\[25\sqrt{5}\]
125
When simplified \[(256) {}^{- ( 4^{- 3/2} )}\] is
8
\[\frac{1}{8}\]
2
\[\frac{1}{2}\]
If \[\frac{3^{2x - 8}}{225} = \frac{5^3}{5^x},\] then x =
2
3
5
4
The value of 64-1/3 (641/3-642/3), is
1
\[\frac{1}{3}\]
-3
-2
If \[\sqrt{5^n} = 125\] then `5nsqrt64`=
25
\[\frac{1}{125}\]
625
\[\frac{1}{5}\]
If (16)2x+3 =(64)x+3, then 42x-2 =
64
256
32
512
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
- \[\frac{1}{2}\]
2
4
\[- \frac{1}{4}\]
If \[\frac{2^{m + n}}{2^{n - m}} = 16\], \[\frac{3^p}{3^n} = 81\] and \[a = 2^{1/10}\],than \[\frac{a^{2m + n - p}}{( a^{m - 2n + 2p} )^{- 1}} =\]
2
- \[\frac{1}{4}\]
9
- \[\frac{1}{8}\]
If \[\frac{3^{5x} \times {81}^2 \times 6561}{3^{2x}} = 3^7\] then x =
3
-3
\[\frac{1}{3}\]
\[- \frac{1}{3}\]
If o <y <x, which statement must be true?
\[\sqrt{x} - \sqrt{y} = \sqrt{x - y}\]
\[\sqrt{x} + \sqrt{x} = \sqrt{2x}\]
\[x\sqrt{y} = y\sqrt{x}\]
\[\sqrt{xy} = \sqrt{x}\sqrt{y}\]
If 10x = 64, what is the value of \[{10}^\frac{x}{2} + 1 ?\]
18
42
80
81
\[\frac{5^{n + 2} - 6 \times 5^{n + 1}}{13 \times 5^n - 2 \times 5^{n + 1}}\] is equal to
\[\frac{5}{3}\]
\[- \frac{5}{3}\]
\[\frac{3}{5}\]
\[- \frac{3}{5}\]
If \[\sqrt{2^n} = 1024,\] then \[{3^2}^\left( \frac{n}{4} - 4 \right) =\]
3
9
27
81
Chapter 2: Exponents of Real Numbers

RD Sharma solutions for Mathematics for Class 9 chapter 2 - Exponents of Real Numbers
RD Sharma solutions for Mathematics for Class 9 chapter 2 (Exponents of Real Numbers) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 solutions in a manner that help students grasp basic concepts better and faster.
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Concepts covered in Mathematics for Class 9 chapter 2 Exponents of Real Numbers are Introduction of Real Number, Concept of Irrational Numbers, Real Numbers and Their Decimal Expansions, Representing Real Numbers on the Number Line, Operations on Real Numbers, Laws of Exponents for Real Numbers.
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