**NCERT Exemplar Problems Class 8 Mathematics Chapter 8 Exponents and Powers**

NCERT Exemplar SolutionsNCERT Solutions MathsRD Sharma Solutions

**Multiple Choice Questions**

**Question. 1 In 2 ^{n}, n is known as**

**(a) base (b) constant**

**(c) exponent (d) variable**

**Solution.**

(c) We know that an is called the nth power of a; and is also read as a raised to the power n.

The rational number a is called the base and n is called the exponent (power or index). In the same way in 2

^{n},n is known as exponent.

**Question. 2 For a fixed base, if the exponent decreases by 1, the number becomes**

** (a) one-tenth of the previous number**

** (b) ten times of the previous number**

** (c) hundredth of the previous number**

** (d) hundred times of the previous number**

** Solution.**

(a) For a fixed base, if the exponent decreases by 1, the number becomes one-tenth of the previous number.

**Question. 3**

**Solution.**

**Question. 4 The value of 1/4 ^{-2} is**

**Solution.**

**Question. 5 The value of 3 ^{5 } Ã· 3^{-6 } is**

**(a) 3**

^{5 }(b) 3^{-6 }(c) 3^{11}(d) 3^{-11}**Solution.**

**Question. 6**

**Solution.**

**Question. 7**

**Solution.**

**Question. 8 The multiplicative inverse of 10 ^{-100 }is**

**(a) 10 (b) 100 (c) 10**

^{100 }(d)10^{-100 }**Solution.**

**Question.9 The value of (-2) ^{2 x 3-1 }is**

**(a) 32 (b) 64 (c) -32 (d) -64**

**Solution.**

**Question.10**

**Solution.**

**Question. 11**

**Solution.**

**Question. 12 If x be any non-zero integer and w, n be negative integers, then x ^{m} x x^{n} is equal to**

**(a) x**

^{m}(b)x^{(m+n)}*(c) x*^{n}*(d) x*^{(m-n)}**Solution.**

**Question. 13 If y be any non-zero integer, then y ^{0} is equal to**

**(a) 1 (b) 0 (c) – 1 (d) not defined**

**Solution.**

**Question.14 If x be any non-zero integer, then _{X}^{-1} is equal to**

**(a) x (b) 1/x (c) – x (d) -1/x**

**Solution.**

**Question. 15**

**Solution.**

**Question. 16**

**Solution.**

**Question. 17**

**Solution.**

**Question. 18**

**Solution.**

**Question. 19**

**Solution.**

**Question. 20**

**Solution.**

**Question. 21**

**Solution.**

**Question. 22**

**Solution.**

**Question. 23**

**Solution.**

**Question.24 The standard form for 0.000064 is**

** (a) 64 x 10 ^{4} (b) 64 x 10^{-4 }(c) 6.4 x 10^{5} (d) 6.4 x 10^{-5}**

**Solution.**

(d) Given, 0.000064 = 0. 64 x 10

^{-4 }=6.4 x 10

^{-5}

Hence, standard form of 0.000064 is 6.4 x 10

^{-5}.

**Question. 25 The standard form for 234000000 is**

** (a) 2.34 x 10 ^{8 } (b) 0.234 x 10^{9 }**

**(c) 2.34 x 10**

^{-8 }(d) 0.234 x 10^{– 9 }**Solution.**

(a) Given, 234000000 = 234 x 10

^{6}= 2.34 x 10

^{+6 }= 2.34 x 10

^{8}

Hence, standard form of 234000000 is 2.34 x 10

^{8}.

**Question.26 The usual form for 2.03 x 10 ^{-5 is }**

**(a) 0.203 (b) 0.00203 (c) 203000 (d) 0.0000203**

**Solution.**

**Question. 27**

**Solution.**

**Question. 28**

**Solution.**

**Question. 29**

**Solution.**

**Question. 30**

**Solution.**

**Question. 31**

**Solution.**

**Question. 32**

**Solution.**

**Question. 33**

**Solution.**

**Fill in the Blanks**

**In questions 34 to 65, fill in the blanks to make the statements true.**

**Question. 34 The multiplicative inverse of 10 ^{10} is_________**

**Solution.**

**Question.35 a ^{3 }x a^{-10}= _________**

**Solution.**

**Question.36 5 ^{0} = _________**

**Solution.**

**Question.37 5 ^{5 }x 5^{-5}= _________**

**Solution.**

**Question.38**

**Solution.**

**Question. 39 The expression for 8 ^{-2 }as a power with the base 2 is_________**

**Solution.**

**Question. 40 Very small numbers can be expressed in standard form by using_________**

** exponents**

** Solution.**

Very small numbers can be expressed in standard form by using negative exponents, i.e. 0.000023 = 2.3 x 10^{-3}

**Question. 41 Very large numbers can be expressed in standard form by using**

** exponents.**

** Solution.**

Very large numbers can be expressed in standard form by using positive exponents,

i.e. 23000 = 23 x 10^{3} =2.3 x 10^{3 }x 10^{1} =2.3 x 10^{4}

**Question. 42 By multiplying (10) ^{5 }by (10)^{-10}, we get**

**Solution.**

**Question.43**

**Solution.**

**Question.44**

**Solution.**

**Question.45**

**Solution.**

**Question.46**

**Solution.**

**Question.47**

**Solution.**

**Question.48**

**Solution.**

**Question.49**

**Solution.**

**Question.50**

**Solution.**

**Question.51**

**Solution.**

**Question.52**

**Solution.**

**Question.53 The value of 3 x 10 ^{-7 } is equal to_______**

**Solution**

Given, 3 x 10

^{-7 }= 3.0 x 10

^{-7 }

Now, placing decimal seven place towards left of original position, we get 0.0000003. Hence, the value of 3 x 10

^{-7 }is equal to 0.0000003.

**Question.54 To add the numbers given in standard form, we first convert them into number with_______exponents.**

** Solution.**

To add the numbers given in standard form, we first convert them into numbers with equal exponents.

e.g. 2.46 x 10^{6 }+ 24.6 x 105 = 2.46 x 10^{5} + 2.46 x 10^{6} = 4.92 x 10^{6}

**Question.55 The standard form for 32500000000 is_______.**

** Solution.**

For standard form, 32500000000 = 3250 x 10^{2} x 10^{2} x 10^{3}

= 3250 x 10^{7} = 3.250 x 10^{10} or 3.25 x 10^{10}

Hence, the standard form for 32500000000 is 3.25 x 10^{10}.

**Question. 56 The standard form for 0.000000008 is_______.**

** Solution.**

For standard form, 0.000000008 = 0.8 x 10^{-8}= 8 x 10^{-9} =8.0 x 10^{-9}

Hence, the standard form for 0.000000008 is 8.0 x 10-9

**Question.57 The usual form for 2.3 x 10 ^{-10 }is_______.**

**Solution.**For usual form, 2.3 x 10

^{-10}= 0.23 x 10

^{-11}

= 0.00000000023

Hence, the usual form for 2.3 x 10

^{-10}is 0.00000000023.

**Question. 58 On dividing 8 ^{5} by_______. we get 8.**

**Solution.**

**Question. 59**

**Solution.**

**Question. 60**

**Solution.**

**Question.61**

**Solution.**

**Question.62**

**Solution.**

**Question.63**

**Solution.**

**Question.64**

**Solution.**

**Question.65**

**Solution.**

**True / False**

**In questions 66 to 90, state whether the given statements are True or False.**

**Question.66**

**Solution.**

**Question.67**

**Solution.**

**Question.68**

**Solution.**

**Question.69 24.58 = 2 x 10 + 4 x 1+5 x 10 + 8 x 100**

** Solution.** False

R H S = 2 x 10+ 4 x 1+ 5 x 10+ 8 x 100=20+ 4 + 50+ 800=874 L H S â‰ R H S

**Question.70 329.25 = 3 x 10 ^{2}+ 2 x 10^{1} + 9 x 10^{0} + 2 x 10^{-1} + 5 x 10^{-2}**

**Solution.**

**Question.71**

**Solution.**

**Question.72**

**Solution.**

**Question.73**

**Solution.**

**Question. 74 5 ^{0} = 5**

**Solution.**

**Question. 75 (-2) ^{0} = 2**

**Solution.**

**Question.76**

**Solution.**

**Question. 77 (-6) ° = – 1**

** Solution.**

**Question. 78**

**Solution.**

**Question. 79**

**Solution.**

**Question. 80**

** Solution.**

**Question. 81**

**Solution.**

**Question. 82**

**Solution.**

**Question. 83**

**Solution.**

**Question. 84**

** Solution.**

**Question.85 The standard form for 0.000037 is 3.7 x 10 ^{-5}**

**Solution.**True

For standard form, 0.000037 = 0.37 x 10

^{-4}= 3.7 x 10

^{-5}

**Question. 86 The standard form for 203000 is 2.03 x 105.**

** Solution.** True

For standard form, 203000 = 203 x 10 x 10 x 10 = 203 x 10^{3}

= 2.03 x 10^{2} x 10^{3}= 2.03 x 10^{5}

**Question. 87 The usual form for 2 x 10 ^{-2} is not equal to 0.02.**

**Solution.**

**Question. 88 The value of 5 ^{-2} is equal to 25.**

**Solution.**False

**Question. 89 Large numbers can be expressed in the standard form by using positive exponents.**

** Solution.**True

e.g. 2360000 = 236 x 10 x 10 x 10 x 10= 236 x 10^{4}

‘ = 2.36 x 10^{4} x 10^{2}=2.36 x 10^{6}

**Question. 90**

**Solution.**

**Question. 91 Solve the following,**

**Solution.**

**Question. 92 Express 3 ^{-5} x 3^{-4} as a power of 3 with positive exponent.**

**Solution.**

**Question. 93 Express 16 ^{-2} as a power with the base 2.**

**Solution.**

**Question. 94**

**Solution.**

**Question. 95**

Solution.

**Question. 96 Express as a power of a rational number with negative exponent.**

**Solution.**

**Question. 97 Find the product of the cube of (-2) and the square of (+4).**

** Solution.**

**Question.98 Simplify**

**Solution.**

**Question. 99 Find the value of x, so that**

**Solution.**

**Question. 100 Divide 293 by 1000000 and express the result in standard form.**

** Solution.**

**Question. 101**

**Solution.**

**Q. 102 By what number should we multiply (-29) °, so that the product becomes (+29) °.**

** Solution.**

**Question. 103 By what number should (-15) ^{-1} be divided so that quotient may be equal to (-15)^{-1 }?**

**Solution.**

**Question.104 Find the multiplicative inverse of (-7) ^{2}Ã· (90)^{-1}**

**Solution.**

**Question.105**

**Solution.**

**Question.106 Write 390000000 in the standard form.**

** Solution.**

**Question. 107 Write 0.000005678 in the standard form.**

** Solution.**

For standard form, 0.000005678 = 0.5678 x 10^{-5}= 5.678 x 10^{-5} x 10^{-1}= 5.678 x 10^{-6 }Hence, 5.678 x 10^{-6} is the standard form of 0.000005678.

**Question.108 Express the product of 3.2 x 10 ^{6} and 4.1 x 10^{1} in the standard form.**

**Solution.**

**Question.109**

**Solution.**

**Question. 110 Some migratory birds travel as much as 15000 km to escape the extreme climatic conditions at home. Write the distance in metres using scientific notation.**

** Solution.**

**Question. 111 Pluto is 5913000000 m from the Sun. Express this in the standard form.**

** Solution.**

**Question. 112 Special balances can weigh something as 0.00000001 gram. Express this number in the standard form.**

** Solution.**

**Question. 113 A sugar factory has annual sales of 3 billion 720 million kilograms of sugar. Express this number in the standard form.**

** Solution.**

**Question. 114 The number of red blood cells per cubic millimetre of blood is approximately mm ^{3})**

**Solution.**The average body contain 5 L of blood.

Also, the number of red blood cells per cubic millimetre of blood is approximately 5.5 million.

Blood contained by body = 5 L = 5 x 100000 mm

^{3}

Red blood cells = 5 x 100000 mm

^{3}

Blood = 5.5 x 1000000 x 5 x 100000= 55 x 5 x 10

^{5 + 5}

= 275 x 10

^{10}= 2.75 x 10

^{10}x 10

^{2}= 2.75 x 10

^{12}

**Question. 115 Express each of the following in standard form:**

**Solution.**

**Question.116**

**Solution.**

**Question.117**

**Solution.**

**In questions 118 and 119, find the value of n.**

**Question.118**

**Solution.**

**Question.119**

**Solution.**

**Question.120**

**Solution.**

**Question.121**

**Solution.**

**Question.122**

**Solution.**

**Question.123 A new born bear weights 4 kg. How many kilograms might a five year old bear weight if its weight increases by the power of 2 in 5 yr?**

** Solution.**

Weight of new born bear = 4 kg

Weight increases by the power of 2 in 5 yr.

Weight of bear in 5 yr = (4)^{2} = 16 kg

**Question.124 The cell of a bacteria doubles in every 30 min. A scientist begins with a single cell. How many cells will be thereafter (a) 12 h (b) 24 h ?**

** Solution.**

**Question.125 Planet A is at a distance of 9.35 x 10 ^{6} km from Earth and planet B is 6.27 x 107 km from Earth. Which planet is nearer to Earth?**

**Solution.**

Distance between planet A and Earth = 9.35 x 10

^{6 }km Distance between planet B and Earth = 6.27 x 10

^{7}km

For finding difference between above two distances, we have to change both in same exponent of 10, i.e. 9.35 x.10

^{6}= 0.935 x 10

^{7}, clearly 6.27 x 10

^{7}is greater.

So, planet A is nearer to Earth.

**Question.126 The cells of a bacteria double itself every hour. How many cells will be there after 8 h, if initially we start with 1 cell. Express the answer in powers.**

** Solution.**

**Question. 127 An insect is on the 0 point of a number line, hopping towards 1. She covers half the distance from her current location to 1 with each hop.**

** So, she will be at 1/2 after one hop, 3/4 after two hops and so on.**

**(a) Make a table showing the insect’s Location for the first 10 hops.**

** (b) Where will the insect be after n hops?**

** (c) Will the insect ever get to 1? Explain.**

** Solution.**

(a) On the basis of given information in the question, we can arrange the following table which shows the insect’s location for the first 10 hops.

**Question. 128 Predicting the ones digit, copy and complete this table and answer the questions that follow.**

**Solution.**

(a) On the basis of given pattern in 1^{x} and 2^{x} , we can make more patterns for 3^{x} 4^{x} , 5^{x} ,6^{x} , 7^{x} , 8^{x} , 9^{x} , 10^{x} .

Thus, we have following table which shows all details about the patterns.

**Question. 129 Astronomy The table shows the mass of the planets, the Sun and the Moon in our solar system.**

**Solution.**

**Question. 130 Investigating Solar System The table shows the average distance from each planet in our solar system to the Sun.**

**Solution.**

**Question. 131 This table shows the mass of one atom for five chemical elements.**

** Use it to answer the question given.**

**(a) Which is the heaviest element?**

** (b) Which element is lighter, Silver or Titanium?**

** (c) List all the five elements in order from lightest to heaviest.**

** Solution.**

**Question. 132 The planet Uranus is approximately 2,896,819,200,000 metres away from the Sun. What is this distance in standard form?**

** Solution.**

Distance between the planet Uranus and the Sun is 2896819200000 m.

Standard form of 2896819200000 = 28968192 x 10 x 10 x 10 x 10 x 10

= 28968192 x 10^{5} = 2.8968192 x 10^{12} m

**Question. 133 An inch is approximately equal to 0.02543 metres. Write this distance in standard form.**

** Solution.** Standard form of 0.02543 m = 0.2543 x 10^{-1} m = 2.543 x 10^{-2} m Hence,’ standard form of 0.025434s 2.543 x 10^{-2} m.

**Question.134 The volume of the Earth is approximately 7.67 x 10 ^{-7} times the volume**

**of the Sun. Express this figure in usual form.**

**Solution.**

**Question.135 An electron’s mass is approximately 9.1093826 x 10 ^{-31} kilograms. What is its mass in grams?**

**Solution.**

**Question. 136 At the end of the 20th century, the world population was approximately 6.1 x 10 ^{9} people. Express this population in usual form. How would you say this number in words?**

**Solution.**

Given, at the end of the 20th century, the world population was 6,1 x 10

^{9}(approx). People population in usual form = 6.1 x 10

^{9}= 6100000000 Hence, population in usual form was six thousand one hundred million.

**Question.137 While studying her family’s history, Shikha discovers records of ancestors 12 generations back. She wonders how many ancestors she had in the past 12 generations. She starts to make a diagram to help her figure this out. The diagram soon becomes very complex**

**Solution.**

(a) On the basis of given diagram, we can make a table that shows the number of ancestors in each of the 12 generations.

**Question. 138 About 230 billion litres of water flows through a river each day, how many litres of water flows through that river in a week? How many litres of water flows through the river in an year? Write your answer in standard notation.**

** Solution.**

**Question. 139 A half-life is the amount of time that it takes for a radioactive substance to decay one-half of its original quantity.**

** Suppose radioactive decay causes 300 grams of a substance to decrease 300 x 2 ^{-3} grams after 3 half-lives. Evaluate 300 x 2^{-3 }to determine how many grams of the substance is left.**

**Explain why the expression 300 x 2**

^{-n}can be used to find the amount of the substance that remains after n half-lives.**Solution.**

**Question. 140 Consider a quantity of a radioactive substance. The fraction of this quantity that remains after t half-lives can be found by using the expression 3 ^{-t}.**

**(a) What fraction of the substance remains after 7 half-lives?**

**(b) After how many half-lives will the fraction be 1/243 of the original?**

**Solution.**

**Question. 141 One fermi is equal to 10 ^{-15} metre. The radius of a proton is 1.3 fermi. Write the radius of a proton (in metres) in standard form.**

**Solution.**The radius of a proton is 1.3 fermi.

One fermi is equal to 10

^{-15}m.

So, the radius of the proton is 1.3 x 10

^{-15}m.

Hence, standard form of radius of the proton is 1.3 x 10

^{-15 }m.

Question. 142 The paper clip below has the indicated length. What is the length in Standard form.

**Solution.**

Length of the paper clip = 0.05 m

In standard form, 0.05 m = 0.5 x 10^{-1} = 5.0 x 10^{-2} m

Hence, the length of the paper clip in standard form is 5.0 x 10^{-2} m

**Question.143 Use the properties of exponents to verify that each statement is true.**

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