Class 8 Maths Chapter 12 Exponents and Powers

Class 8 Maths Chapter 12 Exponents and Powers

Syllabus of Class 8 Mathematics
1.Rational Numbers
2.Linear Equations in one variable
3.Understanding Quadrilaterals
4.Practical Geometry
5.Data Handling
6.Squares Square Roots
7.Cube and Cube Roots
8.Comparing Quantities
9.Algebraic Expressions and Identities
10.Visualising Solid Shapes
11.Mensuration
12.Exponents and Powers
13.Direct and Inverse Properties
14.Factorisation
15.Introduction to Graph
16.Playing with Numbers

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*Class 8 Maths Chapter 12 Exponents and Powers STUDY MATERIAL PDF

Important Formulae

  • Law of Product: am × an = a(m + n)

  • Law of Quotient: am / an = a(m - n)L

  • Lawof Zero Exponent: a0 = 1

  • Law of Negative Exponent: a(-m) = 1 / am

  • Law of Power of a Power: (am)n = a(m * n)

  • Law of Power of a Product: (ab)n = an * bn

Law of Power of a Quotient: (a / b)m = am / bm

Exponents and Powers Class 8 NCERT Solutions (Intext Questions and Exercise)

Question:(i) Find the multiplicative inverse of the following

2^{-4}

Answer:

The detailed explanation for the question is written below,

The multiplicative inverse is \frac{1}{a^{m}}

So, the multiplicative inverse of 2^{-4} is 2^{4}

Question:(ii) Find the multiplicative inverse of the following

10^{-5}

Answer:

Here is the detailed solution for the above question,

As we know,

The multiplicative inverse of a^{m} is \frac{1}{a^{m}}

So, the multiplicative inverse of 10^{-5} is 10^{5}

Question:(iii) Find the multiplicative inverse of the following

7^{-2}

Answer:

The multiplicative inverse of a^{m} is \frac{1}{a^{m}}

So, the multiplicative inverse of 7^{-2} is 7^{2}

Question:(iv) Find the multiplicative inverse of the following

5^{-3}

Answer:

we know,

The multiplicative inverse of a^{m} is \frac{1}{a^{m}}

So, for 5^{-3} multiplicative inverse is 5^{3}

Question:(v) Find the multiplicative inverse of the following

10^{-100}

Answer:

The multiplicative inverse of a^{m} is \frac{1}{a^{m}}

So, the multiplicative inverse of 10^{-100} is 10^{100}

NCERT Solutions for Class 8 Maths Chapter 12 Exponents and Powers Topic 12.2 Powers With Negative Exponents

Question:(i) Expand the following numbers using exponents

 1025.63

Answer:

1025.63 = 1\times 10^{3}+o\times 10^{2}+2\times 10^{1}+5\times 10^{0}+6\times 10^{-1}+3\times 10^{-2}

Question:(ii) Expand the following numbers using exponents

1256.249

Answer:

1256.249

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

NCERT Solutions for Class 8 Maths Chapter 12 Exponents and Powers - Topic 12.3 Laws Of Exponents

Question:1(i) Simplify and write in exponential form

(-2)^{-3}\times (-2)^{-4}

Answer:

this is simplified as follows

(-2)^{-3}\times (-2)^{-4}

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

= \frac{1}{(-2)^{3+4}}=\frac{1}{(-2)^{7}}

= (-2)^{-7}

Question:1(ii) Simplify and write in exponential form

p^3\times p^{-10}

Answer:

this is simplified as follows

p^3\times p^{-10}

p^{3-10} ............. [a^{m}\times a^{n}=a^{m+n}]

=p^{-7}

Question:1(iii) Simplify and write in exponential form

3^2 \times 3^{-4}\times 3^6

Answer:

this can be simplified as follows

3^2 \times 3^{-4}\times 3^6

= 3^{2+(-4)+6} ............. [a^{m}\times a^{n}\times a^{o}=a^{m+n+o}]

= 3^{4}= 81

  Exercise: 12.1

Question:1 (i) Evaluate

3^{-2}

Answer:

The detailed explanation for the above-written question is as follows,

We know that,

a^{-m}=\frac{1}{a^{m}}

So, here m =2 and a = 3

3^{-2}=\frac{1}{3^{2}} = \frac{1}{3}\times \frac{1}{3} = \frac{1}{9}

Question: 1(ii) Evaluate

(-4)^{-2}

Answer:

The detailed explanation for the above-written question is as follows

We know that,

a^{-m}= \frac{1}{a^{m}}

So, here (a = -4) and (m = 2)

Then according to the law of exponent

(-4)^{-2}= \frac{1}{(-4)^{2}} = \frac{1}{(-4)}\times \frac{1}{(-4)} = \frac{1}{16} [ negative \times negative = positive]

Question: 1(iii) Evaluate

\left(\frac{1}{2}\right )^{-5}

Answer:

The detailed solution for the above-written question is as follows

We know that,

(\frac{a}{b})^{m}=\frac{a^{m}}{b^{m}} \& a^{-m} = \frac{1}{a^{m}}

So, here

a = 1 and b = 2 and m =-5

According to the law of exponent

(\frac{1}{2})^{-5}=\frac{1^{-5}}{2^{-5}} = \frac{-1}{2^{-5}}

=(-1)\times -2\times -2\times -2\times -2\times -2 = 32


Question: 2(i) Simplify and express the results in power notation with a positive Exponent.

(-4)^{5}\div (-4)^{8}

Answer:

The detailed solution for the above-written question is as follows

We know the exponential formula

\frac{a^{m}}{a_{n}} = a^{m-n} and a^{-m}= \frac{1}{a^{m}}

So according to this

a = -4, m = 5 and n = 8

\frac{4^{5}}{4_{8}} = -4^{5-8} = -4^{-3}

= (-\frac{1}{4})\times -(\frac{1}{4})\times(-\frac{1}{4}) = -\frac{1}{64}

Question: 2(ii) Simplify and express the results in power notation with a positive Exponent.

\left (\frac{1}{2^3} \right )^2

Answer:

The detailed solution for the above-written question is as follows

We know the exponential formula

\frac{a^{m}}{b^{m}} = (\frac{a}{b})^{m} and a^{-m}= \frac{1}{a^{m}} and (a^{m})^{n} = a^{mn}


So, we have given

a = 1, b=2

By using above exponential law,

\frac{a^{m}}{b^{m}} = \frac{1}{(2^{3})^{2}} = \frac{1}{2^{6}}

= \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} = \frac{1}{64}


Question: 2(iii) Simplify and express the results in power notation with a positive Exponent.

(-3)^4\times \left(\frac{5}{3} \right )^{4}

Answer:

The detailed solution for the above-written question is as follows,

We know the exponential formula

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

So, (-3)^{4}\times (\frac{5}{3})^{4}= (-3)^{4}\times \frac{5^{4}}{3^{4}} = \frac{625}{1}

Question: 2(iv) Simplify and express the results in power notation with a positive Exponent.

(3^{-7}\div 3^{-10})\times 3^{-5}

Answer:

The detailed explanation for the above-written question is as follows

As we know the exponential form

\frac{a^{m}}{b^{n}}= a^{m-n} \& (a^{m}\times a^{n})=a^{m+n}

By using these two form we get,

\frac{3^{-7}}{3^{-10}}\times (3)^{-5}=3^{-7-(-10)} \times (3)^{-5}

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

=\frac{1}{3}\times \frac{1}{3}=\frac{1}{9}

Question:2(v) Simplify and express the results in power notation with a positive Exponent.

2^{-3} \times (-7)^{-3}

Answer:

The detailed solution for the above-written question is as follows,

we know the exponential forms

a^{-m}=\frac{1}{a^{m}} & a^{m}\times b^{m}= (ab)^{m}

So, according to our data,

here initially we use first forms and then the second one.

2^{-3}\times (-7)^{-3}= \frac{1}{2^{3}}\times \frac{1}{(-7)^{3}}

= \frac{1}{(2\times -7)^{3}}= \frac{1}{(-14)^{3}}

Question:3(i) Find the value of

(3^0 + 4^{-1})\times 2^2

Answer:

The detailed explanation for the above-written question is as follows,

As we know that a^{0}=1

So, 3^{0}=1

now,

=(1+\frac{1}{4})\times 2^{2}

=\frac{5}{4}\times 2^{2}\Rightarrow \frac{5}{2^{2}}\times 2^{2}=5/1

Question: 3(ii) Find the value of

(2^{-1}\times 4^{-1})\div 2^{-2}

Answer:

The detailed explanation for the above-written question is as follows

Rewrite the equation

(2^{-1}\times 4^{-1})\div 2^{-2} = (2^{-1}\times 2^{-2})\div 2^{-2}

=(2^{-1+(-2)})\div 2^{-2} ................................. a^{m}\times a^{n}= a^{(m+n)}

=(2^{-3})\div 2^{-2} = 2^{-3-(-2)} ........................ a^{m}\div a^{n}= a^{(m-n)}

= 2^{-1}= \frac{1}{2}

Question: 3(iii) Find the value of

\left (\frac{1}{2}\right)^{-2} + \left (\frac{1}{3}\right)^{-2} + \left (\frac{1}{4}\right)^{-2}

Answer:

The detailed explanation for the above-written question is as follows,

This is the exponential form

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

So, \left (\frac{1}{2}\right)^{-2} + \left (\frac{1}{3}\right)^{-2} + \left (\frac{1}{4}\right)^{-2}

=\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+\frac{1}{4^{-2}} .......................using this form a^{m}= \frac{1}{a^{m}}

=2^{2}+3^{2}+4^{2}

= 4+9+16

= 29

Question: 3(iv) Find the value of

(3^{-1} + 4^{-1} + 5^{-1})^0

Answer:

since we know that

a^{0}=1

(3^{-1} + 4^{-1} + 5^{-1})^0 =1

Question: 3(v) Find the value of

\left \{ \left (\frac{-2}{3} \right )^{-2} \right \}^{2}

Answer:

The detailed explanation for the above-written question is as follows

\left \{ \left (\frac{-2}{3} \right )^{-2} \right \}^{2}

=(-2/3)^{-2\times 2} .............. BY using these form of exponential (a^{m})^{n}=a^{mn}

(-2/3)^{-4}=(-3/2)^{4} ......... use this a^{-m}=\frac{1}{a^{m}}

=\frac{81}{16}

Question:4(i) Evaluate

\frac{8^{-1}\times 5^{3}}{2^{-4}}

Answer:

The detailed explanation for the above written question is as follows

\frac{8^{-1}\times 5^{3}}{2^{-4}}

after rewriting the above equation we get,

\frac{2^{-3}\times 5^{3}}{2^{-4}} =2^{-3-(-4)}\times 5^{3} ...........as we know that \frac{a^{m}}{a^{n}}= a^{m-n}

\\=2^{1}\times 5^{3}\\ = 2\times 125 = 250

An alternate method,

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

here you can use first a^{-m}= \frac{1}{a^{m}} and after that use a^{m}\times a^{n}= a^{m+n}

Question: 4(ii) Evaluate

(5^{-1}\times 2^{-1})\times 6^{-1}

Answer:

The detailed explanation for the above-written question is as follows

We clearly see that this is in the form of a^{-m}=\frac{1}{a^{m}}

So, (5^{-1}\times 2^{-1})\times 6^{-1}= (\frac{1}{5}\times \frac{1}{2})\times \frac{1}{6}

= \frac{1}{60}

Question: 5 Find the value ofm for which

5^m \div 5^{-3} = 5^5

Answer:

We have,

a^{m}\div a^{n}= a^{m-n}

Here a = 5 and n =-3 and m-n = 5

therefore,

5^{m}\div 5^{-3}= 5^{m+3} = 5^{5}

By comparing from both sides we get

m+3 = 5

m= 2

Question: 6(i) Evaluate

\left \{\left (\frac{1}{3} \right )^{-1} - \left( \frac{1}{4} \right )^{-1} \right \}^{-1}

Answer:

The detailed solution for the above-written question is as follows

\left \{\left (\frac{1}{3} \right )^{-1} - \left( \frac{1}{4} \right )^{-1} \right \}^{-1}

= [(1\times 3)-(1\times 4)]^{-1} .............by using a^{-m}= \frac{1}{a^{m}}

= [3-4]^{-1}

\\= [-1]^{-1}\\=-1

Question: 6(ii) Evaluate

\left ( \frac{5}{8} \right )^{-7}\times \left (\frac{8}{5}\right)^{-4}

Answer:

The detailed solution for the above-written question is as follows

\left ( \frac{5}{8} \right )^{-7}\times \left (\frac{8}{5}\right)^{-4}

=\frac{5^{-7}}{8^{-7}}\times \frac{8^{-4}}{5^{-4}} ................ using the form \frac{a^{m}}{b^{m}}= (a/b)^{m}

=5^{-7+4}\times 8^{-4+7} ............using a^{m} \div a^{n} = a^{m-n}

= 5^{-3}\times 8^{+3}

\\= \frac{8^{3}}{5^{3}}\\\\=\frac{512}{125}

Question: 7(i) Simplify

\frac{25\times t^{-4}}{5^{-3}\times 10\times t^{-8}}\;\;(t\neq 0)

Answer:

The detailed solution for the above-written question is as follows

\frac{25\times t^{-4}}{5^{-3}\times 10\times t^{-8}}\;\;(t\neq 0)

we can write 25= 5^{2}

So, after rewriting the equation,

\frac{5^{2}\times t^{-4}}{5^{-3}\times 10\times t^{-8}}

= \frac{5^{2+3}\times t^{-4+8}}{10} .................using the form [a^{m}\div a^{n}= a^{m-n}]

= \frac{5^{5}\times t^{4}}{10} .............(By expanding we have now)

= \frac{625t^{4}}{2}

Question: 7(ii) Simplify

\frac{3^{-5}\times 10^{-5}\times 125}{5^{-7}\times6 ^{-5}}

Answer:

The detailed solution for the above-written question is

\frac{3^{-5}\times 10^{-5}\times 125}{5^{-7}\times6 ^{-5}}

we can write 125 = 5^{3} and 6^{-5} can be written as (2\times 3)^{-5}

Now, rewriting the equation, we get

=\frac{3^{-5}\times 10^{-5}\times 5^{3}}{5^{-7}\times(2\times 3) ^{-5}}

=\frac{3^{-5}\times 10^{-5}\times 5^{3+7}}{(2\times 3) ^{-5}} .............by using [a^{m}\div a^{n}=a^{m-n}]

=\frac{ 10^{-5}\times 5^{10}}{(2) ^{-5}} .....................Use [a^{m}\div a^{n}=a^{m-n}]

5^{10-5}= 5^{5}=3125 .........................As [10^{-5} = (2\times 5)^{-5}=2^{-5}\times 5^{-5}] . 2^{-5} can be cancelled out with the denominator 2^{-5}


Class 8 maths chapter 12 NCERT solutions - Topic 12.4 Use Of Exponents To Express Small Numbers In Standard Form

Question: 1(i) Write the following numbers in expanded form

0.000000564

Answer:

the standard form of 0.000000564 is

\frac{564}{1000000000}=5.64\times10^{-7}

Question: 1(ii) Write the following numbers in expanded form

0.0000021

Answer:

The standard form 0.0000021 is

=\frac{21}{10000000}=2.1\times 10^{-6}

Question: 1(iii) Write the following numbers in expanded form

21600000

Answer:

The standard form 21600000 is

=2.16\times 10^{7}

Question: 1(iv) Write the following numbers in expanded form

15240000

Answer:

The standard form 15240000

=1.524\times 10^{7}

Question: 2 Write the following numbers in expanded form

Answer:

  1. Distance between sun and earth 1.496\times10^{11}m

  2. speed of light is 3\times10^{8}m/s

  3. The avg. diameter of red blood cells is (7\times10^{-6}mm)

  4. the distance of the moon from the earth is (3.84467\times10^{8}m)

  5. size of the plant cell is (1.275\times10^{-5}m)

  6. The diameter of the wire on a computer chip is (3\times10^{-6}m)

  7. the height of the Mount Everest is (8.848\times10^{3}m)

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers - Exercise: 12.2

Question: 1(i) Express the following numbers in standard form

0.0000000000085

Answer:

The standard form is 8.5\times 10^{-12}

Question: 1(ii) Express the following numbers in standard form

0.00000000000942

Answer:

The standard form is 9.42\times 10^{-12}

Question: 1(iii) Express the following numbers in standard form

6020000000000000

Answer:

The standard form is 6.02\times 10^{15}

Question: 1(iv) Express the following numbers in standard form

0.00000000837

Answer:

The standard form of the given number is 8.37\times10^{-9}

Question: 1(v) Express the following numbers in standard form

31860000000

Answer:

The standard form is 3.186\times10^{10}

Question: 2(i) Express the following numbers in usual form.

3.02\times 10^{-6}

Answer:

3.02\times 10^{-6}

=\frac{3.02}{1000000}

=0.00000302

this is the usual form

Question: 2(ii) Express the following numbers in usual form.

4.5 \times 10^4

Answer:

4.5 \times 10^4=4.5\times 10000

=45000

this is the usual form

Question:2(iii) Express the following numbers in usual form.

3\times 10^{-8}

Answer:

3\times 10^{-8}

\frac{3}{100000000} = 0.000000030

this is the usual form

Question: 2(iv) Express the following numbers in usual form.

1.0001\times 10^9

Answer:

1.0001\times 10^9

=1.0001\times 100000000

=1000100000

this is the usual form

Question: 2(v) Express the following numbers in usual form.

5.8\times 10^{12}

Answer:

5.8\times 10^{12}

=5.8\times 100000000000

=5800000000000

this is the usual form

Question:2(vi) Express the following numbers in usual form.

3.61492 \times 10^6

Answer:

3.61492 \times 10^6

= 3.61492\times 1000000

= 3614920

17155

Question:3(i) Express the number appearing in the following statements in standard form.

1 micron is equal to \frac{1}{1000000}m .

Answer:

1 micron is equal to

\frac{1}{1000000}m

= 1\times 10^{-6}

Question:3(ii) Express the number appearing in the following statements in standard form.

Charge of an electron is 0.000,000,000,000,000,000,16 coulomb.

Answer:

Charge of an electron is 0.000,000,000,000,000,000,16 coulomb.

=1.6\times 10^{-19} coulomb.

Question:3(iii) Express the number appearing in the following statements in standard form.

Size of a bacteria is 0.0000005 m.

Answer:

Size of a bacteria is 0.0000005 m

\frac{5}{10000000}=5\times 10^{-7}m

Question:3(iv) Express the number appearing in the following statements in standard form.

Size of a plant cell is 0.00001275 m.

Answer:

Size of a plant cell is0.00001275m

=\frac{1275}{10000}=1.275\times10^{-5}m

Question:3(v) Express the number appearing in the following statements in standard form.

Thickness of a thick paper is 0.07 mm

Answer:

The thickness of a thick paper is 0.07

=\frac{7}{100}=7\times 10^{-2}mm

Question:4 In a stack there are 5 books each of thickness 20 mm and 5 paper sheets each of thickness 0.016 mm.What is the total thickness of the stack.

Answer:

the thickness of each book = 20mm

So, the thickness of 5 books = (5\times 20)=100 mm

the thickness of one paper sheet =0.016mm

So, the thickness of 5 paper sheet = (5\times 0.016)=0.08 mm

the total thickness of the stack = (100+0.08)mm

=100.08 mm or

(1.008\times10^{-2}mm)


EXERCISE: 12.1 (Page 197)
Q1.   Evaluate:
           (i) 3^(-2)
(ii) (-4)^2
(iii) (1/2)^(-5)
Q2.   Simplify and express the result in power notation with positive exponent.
           
           
Q3.   Find the value of:
           
           
           
Q5.   Find the value of m for which 5m ÷ 5–3 = 55.
Sol.    ∵ am ÷ an = am – n
           ∴ L.H.S = 5m ÷ 5–3
           = 5m–(–3) = 5m + 3
           Now 5m + 3 = 55
           Since, bases 5 are equal, therefore exponents are also equal.
           i.e. m + 3 = 5
           or m = 5 – 3 = 2
           Thus, the required value of m is 2.
EXERCISE: 12.2 (Page 200)
Q1.   Express the following numbers in standard form.
           (i) 0.0000000000085
(ii) 0.00000000000942
           (iii) 6020000000000000
(iv) 0.00000000837
           (v) 31860000000
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  1. rsxaxaMarch 12, 2024 at 7:58 AM

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