NCERT Solutions for Class 7 Maths Chapter 11 Exponents and Powers

When someone asks you the speed of light in a vacuum, won't it be difficult to say or write it as 300,000,000 m/s? To make these easier, the concept of exponents and powers can be used. Exponents and powers are one of the important fundamental topics, without which none of the higher mathematics is possible. Exponents can be used to represent very large numbers and small numbers as the power of a base number. The NCERT Solutions of this chapter help students understand important topics like laws of exponents, arithmetic operations on exponents, etc., with a step-by-step approach.

Latest: NCERT Class 7 Maths: Chapterwise Important Formulas with Examples and Points

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1. NCERT Solutions for Class 7 Maths Chapter 11 Exponents and Powers - Important Points
2. NCERT Solutions for Maths Chapter 11 Class 7 - Download PDF
3. NCERT Solutions for Class 7 Maths Exponents and Powers - Exercise
4. Exponents and Powers Class 7 Maths Chapter 11- Topics
5. NCERT Solutions for Class 7 Maths Chapter 11 Exponents and Powers - Points to Remember
6. NCERT Solutions for Class 7 Maths Chapter Wise
7. NCERT Solutions for Class 7 Subject Wise
8. NCERT Books and NCERT Syllabus

These solutions are solved by the experts at Careers360 with reference to the latest syllabus of NCERT Books. These comprehensive step-by-step solutions are of great help during exam preparation. To access the NCERT Solutions for Class 7 Maths for all the chapters of Class 7 Maths, click on the link provided.

NCERT Solutions for Class 7 Maths Chapter 11 Exponents and Powers - Important Points

Exponents: Exponents represent repeated multiplication of a number (base) by itself.

For ${\mathrm{x}}^4$ where x is the base, and 4 is the exponent. It is read as x raised to the power of 4 or the fourth power of $x$.

Understanding Powers of 10:

• Powers of 10 are commonly used in scientific notation and decimal places.
• ${10}^n$ represents a 1 followed by 'n' zeros.

Law of Exponents

$a^m\times a^n=a^{(m+n)}$

$\frac{a^m}{a^n}=a^{(m-n)}$

${\left(a^m\right)}^n=a^{\left(mn\right)}$

$(ab{)}^n=a^n{b}^n$

$(\frac{a}{b}{)}^m=\frac{a^m}{b^m}$

$a^0=1$

$(-{)}^{\text{even number }}=1$

$(-{)}^{\text{odd number }}=-1$

NCERT Solutions for Maths Chapter 11 Class 7 - Download PDF

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NCERT Solutions for Class 7 Maths Exponents and Powers - Exercise

Question: 1(i) Find the value of:

$(i)2^6$

Answer:

The value of $2^6$ is given by

$2\times 2\times 2\times 2\times 2\times 2=64$

Question: 1(ii) Find the value of:

$(ii)9^3$

Answer:

The value of $9^3$ is given by

$9\times 9\times 9=729$

Question: 1(iii) Find the value of:

$(iii){11}^2$

Answer:

The value of ${11}^2$ is given by

$11\times 11=121$

Question: 1(iv) Find the value of:

$(iv)5^4$

Answer:

The value of $5^4$ is given by

$5^4=5\times 5\times 5\times 5=625$

Question: 2 Express the following in exponential form:

$(i)6\times 6\times 6\times 6$

$(ii)t\times t$

$(iii)b\times b\times b\times b$

$(iv)5\times 5\times 7\times 7\times 7$

$(v)2\times 2\times a\times a$

$(vi)a\times a\times a\times c\times c\times c\times c\times d$

Answer:

$(i)6\times 6\times 6\times 6$ can be given as

$6^4$ .

$(ii)t\times t$ can be given as $t^2$ .

$(iii)b\times b\times b\times b$ can be given as $b^4$ .

$(iv)5\times 5\times 7\times 7\times 7$ can be given as

$5^2\times 7^3$ .

$(v)2\times 2\times a\times a$ can be given as

$2^2\times a^2$ .

$(vi)a\times a\times a\times c\times c\times c\times c\times d$ can be given as

$a^3\times c^4\times d$ .

Question: 3 Express each of the following numbers using the exponential notation:

(i) 512 (ii) 343 (iii) 729 (iv) 3125

Answer:

(i) 512

$512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=2^9$

(ii) 343

$343=7\times 7\times 7=7^3$

(iii)729

$729=3\times 3\times 3\times 3\times 3\times 3=3^6$

(iv)3125

$3125=5\times 5\times 5\times 5\times 5=5^5$

Question: 4 Identify the greater number, wherever possible, in each of the following.

$(i)4^{3}or{3}^4$
$(ii)5^{3}or{3}^5$
$(iii)2^{8}or{8}^2$
$(iv){100}^{2}or{2}^{100}$
$(v)2^{10}or{10}^2$

Answer:

$(i)4^{3}or{3}^4$

$4^3=4\times 4\times 4=64$

$3^4=3\times 3\times 3\times 3=81$

since $81>64$

$3^4$ is greater than $4^3$

$(ii)5^{3}or{3}^5$

$5^3=5\times 5\times 5=125$

$3^5=3\times 3\times 3\times 3\times 3=243$

since $243>125$

$3^5$ is greater than $5^3$

$(iii)2^{8}or{8}^2$

$2^8=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256$

$8^2=8\times 8=64$

since $256>64$

$2^8$ is greater than $8^2$

$(iv){100}^{2}or{2}^{100}$

${100}^2=100\times 100=10000$

$2^{100}=2\times 2\times 2\times 2\times 2$ till 100 times 2 since $2^{100}>{100}^2$
$2^{100}$ is greater than ${100}^2$

$(v)2^{10}or{10}^2$

$2^{1}0=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=1024$

${10}^2=10\times 10=100$

since $1024>100$

$2^{1}0$ is greater than ${10}^2$

Question: 5 Express each of the following as a product of powers of their prime factors:

(i) 648 (ii) 405 (iii) 540 (iv) 3,600

Answer:

(i) 648

$648=2^3\times 3^4$

(ii) 405

$405=5\times 3^4$

(iii)540

$540=2^2\times 3^3\times 5$

(iv) 3600

$3600=2^4\times 3^2\times 5^2$

Question: 6(i) Simplify:

$(i)2\times {10}^3$

Answer:

$(i)2\times {10}^3$

can be simplified as

$2\times 10\times 10\times 10=2000$

Question: 6(ii) Simplify:

$(ii)7^2\times 2^2$

Answer:

$(ii)7^2\times 2^2$

can be simplified as

$7\times 7\times 2\times 2=196$

Question: 6(iii) Simplify:

$(iii)2^3\times 5$

Answer:

$(iii)2^3\times 5$

can be simplified as

$2\times 2\times 2\times 5=40$

Question: 6(iv) Simplify:

$(iv)3\times 4^4$

Answer:

$(iv)3\times 4^4$

can be simplified as

$3\times 4\times 4\times 4\times 4=768$

Question: 6(v) Simplify:

$(v)0\times {10}^2$

Answer:

$(v)0\times {10}^2$

can be simplified as

$0\times 10\times 10=0$

Question: 6(vi) Simplify:

$(vi)5^2\times 3^3$

Answer:

$(vi)5^2\times 3^3$

can be simplified as

$5\times 5\times 3\times 3\times 3=675$

Question: 6(vii) Simplify:

$(vii)2^4\times 3^2$

Answer:

$(vii)2^4\times 3^2$

can be simplified as

$2\times 2\times 2\times 2\times 3\times 3=144$

Question: 6(viii) Simplify:

$(viii)3^2\times {10}^4$

Answer:

$(viii)3^2\times {10}^4$

can be simplified as

$3\times 3\times 10\times 10\times 10\times 10=90000$

Question: 7(i) Simplify:

$(i)(-4{)}^3$

Answer:

$(i)(-4{)}^3$

can be simplified as

$-4\times -4\times -4=-64$

Question: 7(ii) Simplify:

$(ii)(-3)\times (-2{)}^3$

Answer:

$(ii)(-3)\times (-2{)}^3$

can be simplified as

$-3\times -2\times -2\times -2=24$

Question: 7(iii) Simplify:

$(iii)(-3{)}^2\times (-5{)}^2$

Answer:

$(iii)(-3{)}^2\times (-5{)}^2$

can be simplified as

$(-3)\times (-3)\times (-5)\times (-5)=225$

Question: 7(iv) Simplify:

$(iv)(-2{)}^3\times (-10{)}^3$

Answer:

$(iv)(-2{)}^3\times (-10{)}^3$

can be simplified as

$(-2)\times (-2)\times (-2)\times -10\times -10\times -10=8000$

Question: 8 Compare the following numbers:

$(i)2.7\times {10}^{12};1.5\times {10}^8$ $(ii)4\times {10}^{14};3\times {10}^{17}$

Answer:

$(i)2.7\times {10}^{12};1.5\times {10}^8$

on comparing exponents of base 10.

$2.7\times {10}^{12}>1.5\times {10}^8$

$(ii)4\times {10}^{14};3\times {10}^{17}$

on comparing exponents of base 10.

$4\times {10}^{14}<3\times {10}^{17}$

Question: 1 Using laws of exponents, simplify and write the answer in exponential form:

$(i)3^2\times 3^4\times 3^8$

$(ii)6^{15}÷6^{10}$

$(iii)a^3\times a^2$

$(iv)7^x\times 7^2$

$(v)(5^2{)}^3÷5^3$

$(vi)2^5\times 5^5$

$(vii)a^4\times b^4$

$(viii)(3^4{)}^3$

$(ix)(2^{20}÷2^{15})\times 2^3$

$(x)8^t÷8^2$

Answer:

$(i)3^2\times 3^4\times 3^8$

can be simplified as $3^{(2+4+8)}=3^{14}$

$(ii)6^{15}÷6^{10}$

can be simplified as $6^{(15-10)}=6^5$

$(iii)a^3\times a^2$

can be simplified as $a^{(3+2)}=a^5$

$(iv)7^x\times 7^2$

can be simplified as $7^{(x+2)}=7^{(x+2)}$

$(v)(5^2{)}^3÷5^3$

can be simplified as
$5^{(2\times 3)}÷5^{(3)}=5^6÷5^3=5^{6-3}=5^3$

$(vi)2^5\times 5^5$

can be simplified as

$(2\times 5{)}^5={10}^5$

$(vii)a^4\times b^4$

can be simplified as $(ab{)}^4$

$(viii)(3^4{)}^3$

can be simplified as $3^{4\times 3}=3^{12}$

$(ix)(2^{20}÷2^{15})\times 2^3$

can be simplified as
$2^{(20-15)}\times 2^3=2^5\times 2^3=2^{(5+3)}=2^8$

$(x)8^t÷8^2$

can be simplified as $8^{(t-2)}$

Question: 2(i) Simplify and express each of the following in exponential form:

$(i)\frac{2^3\times 3^4\times 4}{3\times 32}$

Answer:

$(i)\frac{2^3\times 3^4\times 4}{3\times 32}$

can be simplified as

$=\frac{2^3\times 3^4\times 2^2}{3\times 2^5}$

$=\frac{2^{(3+2)}\times 3^4}{3\times 2^5}$

$=\frac{2^5\times 3^4}{3\times 2^5}$

$=2^{(5-5)}\times 3^{(4-1)}$

$=3^3$

Question: 2(ii) Simplify and express each of the following in exponential form:

$(ii)((5^2{)}^3\times 5^4)÷5^7$

Answer:

$(ii)((5^2{)}^3\times 5^4)÷5^7$

can be simplified as

$=[5^{(2\times 3)}\times 5^4]÷5^7$

$=[5^6\times 5^4]÷5^7$

$=[5^{(6+4)}]÷5^7$

$=[5^{10}]÷5^7$

$=5^{10-7}$

$=5^3$

Question: 2(iii) Simplify and express each of the following in exponential form:

$(iii){25}^4÷5^3$

Answer:

$(iii){25}^4÷5^3$

can be simplified as

$=(5^2{)}^4÷5^3$

$=5^{(2\times 4)}÷5^3$

$=5^8÷5^3$

$=5^{(8-3)}$

$=5^5$

Question: 2(iv) Simplify and express each of the following in exponential form:

$(iv)\frac{3\times 7^2\times {11}^8}{21\times {11}^3}$

Answer:

$(iv)\frac{3\times 7^2\times {11}^8}{21\times {11}^3}$

can be simplified as

$=\frac{3\times 7^2\times {11}^8}{3\times 7\times {11}^3}$

$=3^{(1-1)}\times 7^{(2-1)}\times {11}^{(8-3)}$

$=3^0\times 7^1\times {11}^5$

$=7^1\times {11}^5$

Question: 2(v) Simplify and express each of the following in exponential form:

$(v)\frac{3^7}{3^4\times 3^3}$

Answer:

$(v)\frac{3^7}{3^4\times 3^3}$

can be simplified as

$=\frac{3^7}{3^{4+3}}$

$=\frac{3^7}{3^7}$

$=3^{(7-7)}$

$=3^0$

$=1$

Question: 2(vi) Simplify and express each of the following in exponential form:

$(vi)2^0+3^0+4^0$

Answer:

$(vi)2^0+3^0+4^0$

can be simplified as

$=1+1+1$

$=3$

Question: 2(vii) Simplify and express each of the following in exponential form:

$(vii)2^0\times 3^0\times 4^0$

Answer:

$(vii)2^0\times 3^0\times 4^0$

can be simplified as

$=1\times 1\times 1=1$

Question: 2(viii) Simplify and express each of the following in exponential form:

$(viii)(3^0+2^0)\times 5^0$

Answer:

$(viii)(3^0+2^0)\times 5^0$

can be simplified as

$=(1+1)\times 1$

$=2\times 1$

$=2$

Question: 2(ix) Simplify and express each of the following in exponential form:

$(ix)\frac{2^8\times a^5}{4^3\times a^3}$

Answer:

$(ix)\frac{2^8\times a^5}{4^3\times a^3}$

can be simplified as

$=\frac{2^8\times a^5}{(2^2{)}^3\times a^3}$

$=\frac{2^8\times a^5}{2^6\times a^3}$

$=2^{(8-6)}\times a^{(5-3)}$

$=2^{(2)}\times a^{(2)}$

$=(2a{)}^2$

Question: 2(x) Simplify and express each of the following in exponential form:

$(x)(\frac{a^5}{a^3})\times a^8$

Answer:

$(x)(\frac{a^5}{a^3})\times a^8$

can be simplified as

$=a^{(5-3)}\times a^8$

$=a^{(2)}\times a^8$

$=a^{(2+8)}$

$=a^{(10)}$

Question: 2(xi) Simplify and express each of the following in exponential form:

$(xi)\frac{4^5\times a^8{b}^3}{4^5\times a^5{b}^2}$

Answer:

$(xi)\frac{4^5\times a^8{b}^3}{4^5\times a^5{b}^2}$

can be simplified as

$=4^{(5-5)}\times a^{(8-5)}\times b^{(3-2)}$

$=4^0\times a^3\times b^1$

$=a^3\times b$

Question: 2(xii) Simplify and express each of the following in exponential form:

$(xii)(2^3\times 2{)}^2$

Answer:

$(xii)(2^3\times 2{)}^2$

can be simplified as

$={\left(2^{3+1}\right)}^2$

$=(2^{(4)}{)}^2$

$=2^{(4\times 2)}$

$=2^8$

Question: 3 Say true or false and justify your answer:

$(i)10\times {10}^{11}={100}^{11}$ $(ii)2^3>5^2$ $(iii)2^3\times 3^2=6^5$ $(iv)3^0=(1000{)}^0$

Answer:

$(i)10\times {10}^{11}={100}^{11}$

can be simplified as

$LHS:{10}^{(1+11)}$

$={10}^{(12)}$

Since , $LHS\ne RHS$

Thus, it is false

$(ii)2^3>5^2$

can be simplified as

$LHS=2^3=8$

$RHS=5^2=25$

Since, $LHS\ngtr RHS$

Thus, it is false

$(iii)2^3\times 3^2=6^5$

can be simplified as

$LHS:2^3\times 3^2=8\times 9=72$

$RHS:6^5=7776$

Since , $LHS\ne RHS$

Thus, it is false

$(iv)3^0=(1000{)}^0$

can be simplified as

$LHS:3^0=1$

$RHS:{1000}^0=1$

Since, LHS = RHS

Thus, it is true.

Question: 4 Express each of the following as a product of prime factors only in exponential form:

$(i)108\times 192$ $(ii)270$ $(iii)729\times 64$ $(iv)768$

Answer:

$(i)108\times 192$

$108\times 192=(2^2\times 3^3)\times (2^6\times 3)$

$=2^{(2+6)}\times 3^{(3+1)}$

$=2^8\times 3^4$

$(ii)270$

$270=2\times 3^3\times 5$

$(iii)729\times 64$

$729\times 64=3^6\times 2^6$

$(iv)768$

$768=2^8\times 3$

Question: 5(i) Simplify:

$(i)\frac{(2^5{)}^2\times 7^3}{8^3\times 7}$

Answer:

$(i)\frac{(2^5{)}^2\times 7^3}{8^3\times 7}$

can be simplified as

$=\frac{2^{(5\times 2)}\times 7^3}{(2^3{)}^3\times 7}$

$=\frac{2^{10}\times 7^3}{(2^{(3\times 3)})\times 7}$

$=\frac{2^{10}\times 7^3}{(2^9)\times 7}$

$=2^{(10-9)}\times 7^{(3-1)}$

$=2\times 7^2$

$=2\times 49=98$

Question: 5(ii) Simplify:

$(ii)\frac{25\times 5^2\times t^8}{{10}^3\times t^4}$

Answer:

$(ii)\frac{25\times 5^2\times t^8}{{10}^3\times t^4}$

can be simplified as

$=\frac{5^2\times 5^2\times t^8}{(2\times 5{)}^3\times t^4}$

$=\frac{5^{(2+2)}\times t^8}{2^3\times 5^3\times t^4}$

$=\frac{5^4\times t^8}{2^3\times 5^3\times t^4}$

$=\frac{5^{4-3}\times t^{(8-4)}}{2^3}$

$=\frac{5\times t^4}{2^3}$

$=\frac{5t^4}{8}$

Question: 5(iii) Simplify:

$(iii)\frac{3^5\times {10}^5\times 25}{5^7\times 6^5}$

Answer:

$(iii)\frac{3^5\times {10}^5\times 25}{5^7\times 6^5}$

can be simplified as

$=\frac{3^5\times (2\times 5{)}^5\times 5^2}{5^7\times (2\times 3{)}^5}$

$=\frac{3^5\times 2^5\times 5^5\times 5^2}{5^7\times 2^5\times 3^5}$

$=3^{(5-5)}\times 2^{(5-5)}\times 5^{(5+2-7)}$

$=3^0\times 2^0\times 5^0$

$=1\times 1\times 1=1$

Question: 1 Write the following numbers in their expanded forms:

279404, 3006194, 2806196, 120719, 20068

Answer:

(i) 279404

$279404=200000+70000+9000+400+00+4$

$=2\times 100000+7\times 10000+9\times 1000+4\times 100+0\times 10+4\times 1$

$=2\times {10}^5+7\times {10}^4+9\times {10}^3+4\times {10}^2+0\times {10}^1+4\times {10}^0$

(ii) 3006194

$3006194=3000000+0+0+6000+100+90+4$

$=3\times {10}^6+0\times {10}^5+0\times {10}^4+6\times {10}^3+1\times {10}^2+9\times {10}^1+4\times {10}^0$

(iii) 2806196

$2806196=2000000+800000+0+6000+100+90+6$

$=2\times 1000000+8\times 100000+0\times 10000+6\times 1000+1\times$ $100+9\times 10+6\times 1$

$=2\times {10}^6+8\times {10}^5+0\times {10}^4+6\times {10}^3+1\times {10}^2+9\times {10}^1+6\times {10}^0$

(iv) 120719

$120719=100000+20000+0+700+10+9$

$=1\times 100000+2\times 10000+0\times 1000+7\times 100+1\times 10+9\times 1$ $=1\times {10}^5+2\times {10}^4+0\times {10}^3+7\times {10}^2+1\times {10}^1+9\times {10}^0$

(v) 20068

$20068=20000+0+0+60+8$

$=2\times 10000+0\times 1000+0\times 100+6\times 10+8\times 1$

$=2\times {10}^4+0\times {10}^3+0\times {10}^2+6\times {10}^1+8\times {10}^0$

Question: 2 Find the number from each of the following expanded forms:

$(a)8\times {10}^4+6\times {10}^3+0\times {10}^2+4\times {10}^1+5\times {10}^0$

$(b)4\times {10}^5+5\times {10}^3+3\times {10}^2+2\times {10}^0$

$(c)3\times {10}^4+7\times {10}^2+5\times {10}^0$

$(d)9\times {10}^5+2\times {10}^2+3\times {10}^1$

Answer:

$(a)8\times {10}^4+6\times {10}^3+0\times {10}^2+4\times {10}^1+5\times {10}^0$

$=8\times 10000+6\times 1000+0\times 100+4\times 10+5\times 1$

$=80000+6000+000+40+5$

$=86045$

$(b)4\times {10}^5+5\times {10}^3+3\times {10}^2+2\times {10}^0$

$=4\times 100000+0\times 10000+5\times 1000+3\times 100+0\times 10+2\times 1$

$=400000+00000+5000+300+00+2$

$=405302$

$(c)3\times {10}^4+7\times {10}^2+5\times {10}^0$

$=3\times 10000+0\times 1000+7\times 100+0\times 10+5\times 1$

$=30000+0000+700+00+5$

$=30705$

$(d)9\times {10}^5+2\times {10}^2+3\times {10}^1$

$=9\times 100000+0\times 10000+0\times 1000+2\times 100+3\times 10+0\times 1$

$=900000+00000+0000+200+30+0$

$=900230$

Thus, the above problems are simplified in simpler forms.

Question: 3 Express the following numbers in standard form:

(i) 5,00,00,000 (ii) 70,00,000 (iii) 3,18,65,00,000 (iv) 3,90,878 (v) 39087.8 (vi) 3908.78

Answer:

(i) 5,00,00,000

$50000000=5\times 10000000=5\times {10}^7$

(ii) 70,00,000

$7000000=7\times 1000000=7\times {10}^6$

(iii) 3,18,65,00,000

$3186500000=31865\times 100000$

$=3.1865\times 10000\times 100000$

$=3.1865\times {10}^9$

(iv) 3,90,878

$=3.90878\times 100000$

$=3.90878\times {10}^5$

(v) 39087.8

$=3.90878\times 10000$