# NCERT solutions for Mathematics Exemplar Class 8 chapter 8 - Exponents and Powers [Latest edition]

## Chapter 8: Exponents and Powers

Exercise
Exercise [Pages 249 - 274]

### NCERT solutions for Mathematics Exemplar Class 8 Chapter 8 Exponents and Powers Exercise [Pages 249 - 274]

#### Choose the correct alternative:

Exercise | Q 1 | Page 249

In 2n, n is known as ______.

• Base

• Constant

• Exponent

• Variable

Exercise | Q 2 | Page 249

For a fixed base, if the exponent decreases by 1, the number becomes ______.

• One-tenth of the previous number

• Ten times of the previous number

• Hundredth of the previous number

• Hundred times of the previous number

Exercise | Q 3 | Page 249

3–2 can be written as ______.

• 32

• 1/3^2

• 1/3^-2

• - 2/3

Exercise | Q 4 | Page 249

The value of 1/4^-2 is ______.

• 16

• 8

• 1/16

• 1/8

Exercise | Q 5 | Page 250

The value of 35 ÷ 3–6 is ______.

• 35

• 3-6

• 311

• 3-11

Exercise | Q 6 | Page 250

The value of (2/5)^-2 is ______.

• 4/5

• 4/25

• 25/4

• 5/2

Exercise | Q 7 | Page 250

The reciprocal of (2/5)^-1 is ______.

• 2/5

• 5/2

• - 5/2

•  - 2/5

Exercise | Q 8 | Page 250

The multiplicative inverse of 10–100 is ______.

• 10

• 100

• 10100

• 10–100

Exercise | Q 9 | Page 250

The value of (–2)2×3 –1 is ______.

• 32

• 64

• – 32

• – 64

Exercise | Q 10 | Page 250

The value of (- 2/3)^4 is equal to ______.

• 16/81

• 81/16

• (-16)/81

• 81/(-16)

Exercise | Q 11 | Page 251

The multiplicative inverse of (- 5/9)^99 is ______.

• (- 5/9)^99

• (5/9)^99

• (9/(-5))^99

• (9/5)^99

Exercise | Q 12 | Page 251

If x be any non-zero integer and m, n be negative integers, then xm × xn is equal to ______.

• x

• xm+n

• xn

• xm–n

Exercise | Q 13 | Page 251

If y be any non-zero integer, then y0 is equal to ______.

• 1

• 0

• – 1

• Not defined

Exercise | Q 14 | Page 251

If x be any non-zero integer, then x–1 is equal to ______.

• x

• 1/x

• – x

• (-1)/x

Exercise | Q 15 | Page 251

If x be any integer different from zero and m be any positive integer, then x–m is equal to ______.

• xm

• –xm

• 1/x^m

• (-1)/x^m

Exercise | Q 16 | Page 251

If x be any integer different from zero and m, n be any integers, then (xm)n is equal to ______.

• x^(m + n)

• x^(mn)

• x^(m/n)

• x^(m - n)

Exercise | Q 17 | Page 251

Which of the following is equal to (- 3/4)^-3?

• (3/4)^-3

• -(3/4)^-3

• (4/3)^3

• (- 4/3)^3

Exercise | Q 18 | Page 251

(- 5/7)^-5 is equal to ______.

• (5/7^-5

• (5/7)^5

• (7/5)^5

• ((-7)/5)^5

Exercise | Q 19 | Page 252

((-7)/5)^-1 is equal to ______.

• 5/7

• - 5/7

• 7/5

• (-7)/5

Exercise | Q 20 | Page 252

(–9)3 ÷ (–9)8 is equal to ______.

• (9)5

• (9)–5

• (– 9)5

• (– 9)–5

Exercise | Q 21 | Page 252

For a non-zero integer x, x7 ÷ x12 is equal to ______.

• x5

• x19

• x–5

• x–19

Exercise | Q 22 | Page 252

For a non-zero integer x, (x4)–3 is equal to ______.

• x12

• x–12

• x64

• x–64

Exercise | Q 23 | Page 252

The value of (7–1 – 8–1) –1 – (3–1 – 4–1) –1 is ______.

• 44

• 56

• 68

• 12

Exercise | Q 24 | Page 252

The standard form for 0.000064 is ______.

• 64 × 104

• 64 × 10–4

• 6.4 × 105

• 6.4 × 10–5

Exercise | Q 25 | Page 252

The standard form for 234000000 is ______.

• 2.34 × 108

• 0.234 × 109

• 2.34 × 10–8

• 0.234 × 10–9

Exercise | Q 26 | Page 252

The usual form for 2.03 × 10–5 is ______.

• 0.203

• 0.00203

• 203000

• 0.0000203

Exercise | Q 27 | Page 253

(1/10)^0 is equal to ______.

• 0

• 1/10

• 1

• 10

Exercise | Q 28 | Page 253

(3/4)^5 ÷ (5/3)^5 is equal to ______.

• (3/4 ÷ 5/3)^5

• (3/4 ÷ 5/3)^1

• (3/4 ÷ 5/3)^0

• (3/4 ÷ 5/3)^10

Exercise | Q 29 | Page 253

For any two non-zero rational numbers x and y, x4 ÷ y4 is equal to ______.

• (x ÷ y)0

• (x ÷ y)1

• (x ÷ y)4

• (x ÷ y)8

Exercise | Q 30 | Page 253

For a non-zero rational number p, p13 ÷ p8 is equal to ______.

• p5

• p21

• p–5

• p–19

Exercise | Q 31 | Page 253

For a non-zero rational number z, (z–2)3 is equal to ______.

• z6

• z–6

• z1

• z4

Exercise | Q 32 | Page 253

Cube of - 1/2 is ______.

• 1/8

• 1/16

• - 1/8

• - 1/16

Exercise | Q 33 | Page 253

Which of the following is not the reciprocal of (2/3)^4?

• (3/2)^4

• (3/2)^-4

• (2/3)^-4

• 3^4/2^4

#### Fill in the blanks:

Exercise | Q 34 | Page 253

The multiplicative inverse of 1010 is ______.

Exercise | Q 35 | Page 253

a3 × a–10 = ______.

Exercise | Q 36 | Page 254

50 = ______.

Exercise | Q 37 | Page 254

55 × 5–5 = ______.

Exercise | Q 38 | Page 254

The value of (1/2^3)^2 is equal to ______.

Exercise | Q 39 | Page 254

The expression for 8–2 as a power with the base 2 is ______.

Exercise | Q 40 | Page 254

Very small numbers can be expressed in standard form by using ______ exponents.

Exercise | Q 41 | Page 254

Very large numbers can be expressed in standard form by using ______ exponents.

Exercise | Q 42 | Page 254

By multiplying (10)5 by (10)–10 we get ______.

Exercise | Q 43 | Page 254

[(2/13)^-6 ÷ (2/13)^3]^3 xx (2/13)^-9 = ______.

Exercise | Q 44 | Page 254

Find the value [4–1 + 3–1 + 6–2]–1 = ______.

Exercise | Q 45 | Page 254

[2–1 + 3–1 + 4–1]0 = ______.

Exercise | Q 46 | Page 254

The standard form of (1/100000000) is ______.

Exercise | Q 47 | Page 254

The standard form of 12340000 is ______.

Exercise | Q 48 | Page 254

The usual form of 3.41 × 106 is ______.

Exercise | Q 49 | Page 254

The usual form of 2.39461 × 106 is _______.

Exercise | Q 50 | Page 254

If 36 = 6 × 6 = 62, then 1/36 expressed as a power with the base 6 is ______.

Exercise | Q 51 | Page 255

By multiplying (5/3)^4 by ______ we get 5^4.

Exercise | Q 52 | Page 255

35 ÷ 3–6 can be simplified as ______.

Exercise | Q 53 | Page 255

The value of 3 × 10–7 is equal to ______.

Exercise | Q 54 | Page 255

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

Exercise | Q 55 | Page 255

The standard form for 32,50,00,00,000 is ______.

Exercise | Q 56 | Page 255

The standard form for 0.000000008 is ______.

Exercise | Q 57 | Page 255

The usual form for 2.3 × 10-10 is ______.

Exercise | Q 58 | Page 255

On dividing 85 by ______ we get 8.

Exercise | Q 59 | Page 255

On multiplying ______ by 2–5 we get 25.

Exercise | Q 59 | Page 255

On multiplying ______ by 2–5 we get 25.

Exercise | Q 60 | Page 255

The value of [3–1 × 4–1]2 is ______.

Exercise | Q 61 | Page 255

The value of [2–1 × 3–1]–1 is ______.

Exercise | Q 62 | Page 255

By solving (60 – 70) × (60 + 70) we get ______.

Exercise | Q 63 | Page 255

The expression for 35 with a negative exponent is ______.

Exercise | Q 64 | Page 255

The value for (–7)6 ÷ 76 is ______.

Exercise | Q 65 | Page 255

The value of [1–2 + 2–2  + 3–2 ] × 62 is ______.

#### State whether the following statement is True or False:

Exercise | Q 66 | Page 255

The multiplicative inverse of (– 4)–2 is (4)–2.

• True

• False

Exercise | Q 67 | Page 255

The multiplicative inverse of (3/2)^2 is not equal to (2/3)^-2.

• True

• False

Exercise | Q 68 | Page 255

10^-2 = 1/100

• True

• False

Exercise | Q 69 | Page 255

24.58 = 2 × 10 + 4 × 1 + 5 × 10 + 8 × 100

• True

• False

Exercise | Q 70 | Page 255

329.25 = 3 × 102 + 2 × 101 + 9 × 100 + 2 × 10–1 + 5 × 10–2

• True

• False

Exercise | Q 71 | Page 255

(–5)–2 × (–5)–3 = (–5)–6

• True

• False

Exercise | Q 72 | Page 255

(–4)–4 × (4)–1 = (4)5

• True

• False

Exercise | Q 73 | Page 256

(2/3)^-2 xx (2/3)^-5 = (2/3)^10

• True

• False

Exercise | Q 74 | Page 256

50 = 5

• True

• False

Exercise | Q 75 | Page 256

(–2)0 = 2

• True

• False

Exercise | Q 76 | Page 256

(- 8/2)^0 = 0

• True

• False

Exercise | Q 77 | Page 256

(–6)0 = –1

• True

• False

Exercise | Q 78 | Page 256

(–7)–4 × (–7)2 = (–7)–2

• True

• False

Exercise | Q 79 | Page 256

The value of 1/4^-2 is equal to 16.

• True

• False

Exercise | Q 80 | Page 256

The expression for 4–3 as a power with the base 2 is 26.

• True

• False

Exercise | Q 81 | Page 256

ap × bq = (ab)pq

• True

• False

Exercise | Q 82 | Page 256

x^m/y^m = (y/x)^-m

• True

• False

Exercise | Q 83 | Page 256

a^m = 1/a^-m

• True

• False

Exercise | Q 84 | Page 256

The exponential form for (-2)^4 xx (5/2)^4 is 54.

• True

• False

Exercise | Q 85 | Page 256

The standard form for 0.000037 is 3.7 × 10–5.

• True

• False

Exercise | Q 86 | Page 257

The standard form for 203000 is 2.03 × 105.

• True

• False

Exercise | Q 87 | Page 257

The usual form for 2 × 10–2 is not equal to 0.02.

• True

• False

Exercise | Q 88 | Page 257

The value of 5–2 is equal to 25.

• True

• False

Exercise | Q 89 | Page 257

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

• True

• False

Exercise | Q 90 | Page 257

am × bm = (ab)m

• True

• False

Exercise | Q 91.(i) | Page 257

Solve the following:

100^-10

Exercise | Q 91.(ii) | Page 257

Solve the following:

2^-2 xx 2^-3

Exercise | Q 91.(iii) | Page 257

Solve the following:

(1/2)^-2 + (1/2)^-3

Exercise | Q 92 | Page 257

Express 3–5 × 3–4 as a power of 3 with positive exponent.

Exercise | Q 93 | Page 257

Express 16–2 as a power with the base 2.

Exercise | Q 94 | Page 257

Express 27/64 and (-27)/64 as powers of a rational number.

Exercise | Q 95 | Page 257

Express 16/81 and (-16)/81 as powers of a rational number.

Exercise | Q 96.(a) | Page 257

Express as a power of a rational number with negative exponent.

(((-3)/2)^-2)^-3

Exercise | Q 96.(b) | Page 257

Express as a power of a rational number with negative exponent.

(2^5 ÷ 2^8) xx 2^-7

Exercise | Q 97 | Page 257

Find the product of the cube of (–2) and the square of (+4).

Exercise | Q 98.(i) | Page 257

Simplify:

(1/4)^-2 + (1/2)^-2 + (1/3)^-2

Exercise | Q 98.(ii) | Page 257

Simplify:

(((-2)/3)^-2)^3 xx (1/3)^-4 xx 3^-1 xx 1/6

Exercise | Q 98.(iii) | Page 257

Simplify:

(49 xx z^-3)/(7^-3 xx 10 xx z^-5) (z ≠ 0)

Exercise | Q 98.(iv) | Page 257

Simplify:

(2^5 ÷ 2^8) xx 2^-7

Exercise | Q 99.(i) | Page 258

Find the value of x so that (5/3)^-2 xx (5/3)^-14 = (5/3)^(8x)

Exercise | Q 99.(ii) | Page 258

Find the value of x so that (-2)^3 xx (-2)^-6 = (-2)^(2x - 1)

Exercise | Q 99.(iii) | Page 258

Find the value of x so that (2^-1 + 4^-1 + 6^-1 + 8^-1)^x = 1

Exercise | Q 100 | Page 258

Divide 293 by 10,00,000 and express the result in standard form

Exercise | Q 101 | Page 258

Find the value of x–3 if x = (100)1–4 ÷ (100)0.

Exercise | Q 102 | Page 258

By what number should we multiply (–29)0 so that the product becomes (+29)0.

Exercise | Q 103 | Page 258

By what number should (–15)–1 be divided so that quotient may be equal to (–15)–1?

Exercise | Q 104 | Page 259

Find the multiplicative inverse of (–7)–2 ÷ (90)–1.

Exercise | Q 105 | Page 259

If 5^(3x–1) ÷ 25 = 125, find the value of x.

Exercise | Q 106 | Page 259

Write 39,00,00,000 in the standard form.

Exercise | Q 107 | Page 259

Write 0.000005678 in the standard form.

Exercise | Q 108 | Page 259

Express the product of 3.2 × 106 and 4.1 × 10–1 in the standard form.

Exercise | Q 109 | Page 259

Express (1.5 xx 10^6)/(2.5 xx 10^-4) in the standard form.

Exercise | Q 110 | Page 259

Some migratory birds travel as much as 15,000 km to escape the extreme climatic conditions at home. Write the distance in metres using scientific notation.

Exercise | Q 111 | Page 259

Pluto is 59,1,30,00, 000 m from the sun. Express this in the standard form

Exercise | Q 112 | Page 259

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

Exercise | Q 113 | Page 259

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

Exercise | Q 114 | Page 259

The number of red blood cells per cubic millimetre of blood is approximately 5.5 million. If the average body contains 5 litres of blood, what is the total number of red cells in the body? Write the standard form. (1 litre = 1,00,000 mm3)

Exercise | Q 115.(a) | Page 259

Express the following in standard form:

The mass of a proton in gram is 1673/1000000000000000000000000000

Exercise | Q 115.(b) | Page 259

Express the following in standard form:

A Helium atom has a diameter of 0.000000022 cm.

Exercise | Q 115.(c) | Page 259

Express the following in standard form:

Mass of a molecule of hydrogen gas is about 0.00000000000000000000334 tons.

Exercise | Q 115.(d) | Page 259

Express the following in standard form:

Human body has 1 trillon of cells which vary in shapes and sizes.

Exercise | Q 115.(e) | Page 259

Express the following in standard form:

Express 56 km in m.

Exercise | Q 115.(f) | Page 259

Express the following in standard form:

Express 5 tons in g.

Exercise | Q 115.(g) | Page 259

Express the following in standard form:

Express 2 years in seconds.

Exercise | Q 115.(h) | Page 259

Express the following in standard form:

Express 5 hectares in cm2 (1 hectare = 10000 m2)

Exercise | Q 116 | Page 260

Find x so that (2/9)^3 xx (2/9)^-6 xx (2/9)^(2x - 1)

Exercise | Q 117 | Page 260

By what number should ((-3)/2)^3 be divided so that the quotient may be (4/27)^-2?

Exercise | Q 118 | Page 260

Find the value of n.

6^n/6^-2 = 6^3

Exercise | Q 119 | Page 260

Find the value of n.

(2^n xx 2^6)/2^-3 = 2^18

Exercise | Q 120 | Page 260

(125 xx x^-3)/(5^-3 xx 25 xx x^-6)

Exercise | Q 121 | Page 260

(16 xx 10^2 xx 64)/(2^4 xx 4^2)

Exercise | Q 122 | Page 260

If (5^m xx 5^3 xx 5^-2)/5^-5, then find m.

Exercise | Q 123 | Page 260

A new born bear weighs 4 kg. How many kilograms might a five-year-old bear weigh if its weight increases by the power of 2 in 5 years?

Exercise | Q 124.(a) | Page 260

The cells of a bacteria double in every 30 minutes. A scientist begins with a single cell. How many cells will be there after 12 hours?

Exercise | Q 124.(b) | Page 260

The cells of a bacteria double in every 30 minutes. A scientist begins with a single cell. How many cells will be there after 24 hours?

Exercise | Q 125 | Page 260

Planet A is at a distance of 9.35 × 106 km from Earth and planet B is 6.27 × 107 km from Earth. Which planet is nearer to Earth?

Exercise | Q 126 | Page 260

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

Exercise | Q 127.(a) | Page 261

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.

Make a table showing the insect’s location for the first 10 hops.

Exercise | Q 127.(b) | Page 261

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.

Where will the insect be after n hops?

Exercise | Q 127.(c) | Page 261

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.

Will the insect ever get to 1? Explain.

Exercise | Q 128 | Page 261

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

 Powers Table x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x 1 1 2 2 1 4 3 1 8 4 1 16 5 1 32 6 1 64 7 1 128 8 1 256 Ones Digits of thePowers 1 2, 4, 8, 6

(a) Describe patterns you see in the ones digits of the powers.
(b) Predict the ones digit in the following:
(i) 412
(ii) 920
(iii) 317
(iv) 5100
(v) 10500
(c) Predict the ones digit in the following:
(i) 3110
(ii) 1210
(iii) 1721
(iv) 2910

Exercise | Q 129 | Page 262

Astronomy The table shows the mass of the planets, the sun and the moon in our solar system.

 Celestial Body Mass (kg) Mass (kg) Standard Notation Sun 1,990,000,000,000,000,000,000,000,000,000 1.99 × 1030 Mercury 330,000,000,000,000,000,000,000 Venus 4,870,000,000,000,000,000,000,000 Earth 5,970,000,000,000,000,000,000,000 Mars 642,000,000,000,000,000,000,000,000,000 Jupiter 1,900,000,000,000,000,000,000,000,000 Saturn 568,000,000,000,000,000,000,000,000 Uranus 86,800,000,000,000,000,000,000,000 Neptune 102,000,000,000,000,000,000,000,000 Pluto 12,700,000,000,000,000,000,000 Moon 73,500,000,000,000,000,000,000

(a) Write the mass of each planet and the Moon in scientific notation.
(b) Order the planets and the moon by mass, from least to greatest.
(c) Which planet has about the same mass as earth?

Exercise | Q 130 | Page 262

Investigating Solar System The table shows the average distance from each planet in our solar system to the sun.

 Planet Distance from Sun (km) Distance from Sun (km) StandardNotation Earth 149,600,000 Jupiter 778,300,000 Mars 227,900,000 Mercury 57,900,000 Neptune 4,497,000,000 Pluto 5,900,000,000 Saturn 1,427,000,000 Uranus 2,870,000,000 Venus 108,200,000

(a) Complete the table by expressing the distance from each planet to the Sun in scientific notation.
(b) Order the planets from closest to the sun to farthest from the sun

Exercise | Q 131 | Page 262

This table shows the mass of one atom for five chemical elements. Use it to answer the question given.

 Element Mass of atom (kg) Titanium 7.95 × 10–26 Lead 3.44 × 10–25 Silver 1.79 × 10–25 Lithium 1.15 × 10–26 Hydrogen 1.674 × 10–27

(a) Which is the heaviest element?
(b) Which element is lighter, Silver or Titanium?
(c) List all five elements in order from lightest to heaviest.

Exercise | Q 132 | Page 263

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

Exercise | Q 133 | Page 263

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

Exercise | Q 134 | Page 263

The volume of the Earth is approximately 7.67 × 10–7 times the volume of the Sun. Express this figure in usual form.

Exercise | Q 135 | Page 263

An electron’s mass is approximately 9.1093826 × 10–31 kilograms. What is this mass in grams?

Exercise | Q 136 | Page 263

At the end of the 20th century, the world population was approximately 6.1 × 109 people. Express this population in usual form. How would you say this number in words?

Exercise | Q 137 | Page 263

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

(a) Make a table and a graph showing the number of ancestors in each of the 12 generations.
(b) Write an equation for the number of ancestors in a given generation n

Exercise | Q 138 | Page 264

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.

Exercise | Q 139 | Page 264

A half-life is the amount of time that it takes for a radioactive substance to decay to one half of its original quantity. Suppose radioactive decay causes 300 grams of a substance to decrease to 300 × 2–3 grams after 3 half-lives. Evaluate 300 × 2–3 to determine how many grams of the substance are left.

Exercise | Q 140.(a) | Page 264

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. What fraction of substance remains after 7 half-lives?

Exercise | Q 140.(b) | Page 264

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. After how many half-lives will the fraction be 1/243 of the original?

Exercise | Q 141 | Page 264

One Fermi is equal to 10–15 metre. The radius of a proton is 1.3 Fermis. Write the radius of a proton in metres in standard form.

Exercise | Q 142 | Page 264

The paper clip below has the indicated length. What is the length in standard form

Exercise | Q 143.(a) | Page 264

Use the properties of exponents to verify that statement is true.

1/4 (2^n) = 2^(n - 2)

Exercise | Q 143.(b) | Page 264

Use the properties of exponents to verify that statement is true.

4^(n - 1) = 1/4(4)^n

Exercise | Q 143.(c) | Page 264

Use the properties of exponents to verify that statement is true.

25(5^(n - 2)) = 5^n

Exercise | Q 144 | Page 264

Fill in the blank

Exercise | Q 145 | Page 264

There are 864,00 seconds in a day. How many days long is a second? Express your answer in scientific notation.

Exercise | Q 146 | Page 265

The given table shows the crop production of a State in the year 2008 and 2009. Observe the table given below and answer the given questions.

 Crop 2008 Harvest(Hectare) Increase/Decrease(Hectare) in 2009 Bajra 1.4 × 103 – 100 Jowar 1.7 × 106 – 440,000 Rice 3.7 × 103 – 100 Wheat 5.1 × 105 + 190,000

(a) For which crop(s) did the production decrease?
(b) Write the production of all the crops in 2009 in their standard form.
(c) Assuming the same decrease in rice production each year as in 2009, how many acres will be harvested in 2015? Write in standard form.

#### Stretching Machine

Exercise | Q 147 | Page 265

Suppose you have a stretching machine which could stretch almost anything. For example, if you put a 5 metre stick into a (× 4) stretching machine (as shown below), you get a 20 metre stick. Now if you put 10 cm carrot into a (× 4) machine, how long will it be when it comes out?

Exercise | Q 148.(a) | Page 265

Two machines can be hooked together. When something is sent through this hook up, the output from the first machine becomes the input for the second. Which two machines hooked together do the same work a (× 102) machine does? Is there more than one arrangement of two machines that will work?

Exercise | Q 148.(b) | Page 265

Two machines can be hooked together. When something is sent through this hook up, the output from the first machine becomes the input for the second. Which stretching machine does the same work as two (× 2) machines hooked together?

#### Repeater Machine

Exercise | Q 149 | Page 266

Similarly, repeater machine is a hypothetical machine which automatically enlarges items several times. For example, sending a piece of wire through a (× 24) machine is the same as putting it through a (× 2) machine four times. So, if you send a 3 cm piece of wire through a (× 24) machine, its length becomes 3 × 2 × 2 × 2 × 2 = 48 cm. It can also be written that a base (2) machine is being applied 4 times.

What will be the new length of a 4 cm strip inserted in the machine?

Exercise | Q 150 | Page 266

For the following repeater machines, how many times the base machine is applied and how much the total stretch is?

Exercise | Q 151 | Page 266

Find three repeater machines that will do the same work as a (× 64) machine. Draw them, or describe them using exponents.

Exercise | Q 152 | Page 266

What will the following machine do to a 2 cm long piece of chalk?

Exercise | Q 153.(a) | Page 267

In a repeater machine with 0 as an exponent, the base machine is applied 0 times. What do these machines do to a piece of chalk?

Exercise | Q 153.(b) | Page 266

In a repeater machine with 0 as an exponent, the base machine is applied 0 times. What do you think the value of 60 is?

#### Shrinking Machine

Exercise | Q 154 | Page 268

In a shrinking machine, a piece of stick is compressed to reduce its length. If 9 cm long sandwich is put into the shrinking machine below, how many cm long will it be when it emerges?

Exercise | Q 155.(i) | Page 268

What happens when 1 cm worms are sent through these hook-ups?

Exercise | Q 155.(ii) | Page 268

What happens when 1 cm worms are sent through these hook-ups?

Exercise | Q 156 | Page 268

Sanchay put a 1 cm stick of gum through a (1 × 3–2) machine. How long was the stick when it came out?

Exercise | Q 157 | Page 268

Ajay had a 1cm piece of gum. He put it through repeater machine given below and it came out 1/(100,000) cm long. What is the missing value?

Exercise | Q 158.(a) | Page 268

Find a single machine that will do the same job as the given hook-up.

a (× 23) machine followed by (× 2–2) machine.

Exercise | Q 158.(b) | Page 268

Find a single machine that will do the same job as the given hook-up.

a (× 24) machine followed by (xx (1/2)^2) machine.

Exercise | Q 158.(c) | Page 268

Find a single machine that will do the same job as the given hook-up.

a (× 599) machine followed by a (5–100) machine.

Exercise | Q 159.(a) | Page 269

Find a single repeater machine that will do the same work as hook-up.

Exercise | Q 159.(b) | Page 269

Find a single repeater machine that will do the same work as hook-up.

Exercise | Q 159.(c) | Page 269

Find a single repeater machine that will do the same work as hook-up.

Exercise | Q 159.(d) | Page 269

Find a single repeater machine that will do the same work as hook-up.

Exercise | Q 159.(e) | Page 269

Find a single repeater machine that will do the same work as hook-up.

Exercise | Q 159.(f) | Page 269

Find a single repeater machine that will do the same work as hook-up.

Exercise | Q 160.(a) | Page 270

For hook-up, determine whether there is a single repeater machine that will do the same work. If so, describe or draw it.

Exercise | Q 160.(b) | Page 270

For hook-up, determine whether there is a single repeater machine that will do the same work. If so, describe or draw it.

Exercise | Q 160.(c) | Page 270

For hook-up, determine whether there is a single repeater machine that will do the same work. If so, describe or draw it.

Exercise | Q 160.(d) | Page 270

For hook-up, determine whether there is a single repeater machine that will do the same work. If so, describe or draw it.

Exercise | Q 160.(e) | Page 270

For hook-up, determine whether there is a single repeater machine that will do the same work. If so, describe or draw it.

Exercise | Q 161 | Page 271

Shikha has an order from a golf course designer to put palm trees through a (× 23) machine and then through a (× 33) machine. She thinks she can do the job with a single repeater machine. What single repeater machine should she use?

Exercise | Q 162 | Page 271

Neha needs to stretch some sticks to 252 times their original lengths, but her (× 25) machine is broken. Find a hook-up of two repeater machines that will do the same work as a (× 252) machine. To get started, think about the hookup you could use to replace the (× 25) machine.

Exercise | Q 163.(a) | Page 271

Supply the missing information for diagram.

Exercise | Q 163.(b) | Page 271

Supply the missing information for diagram.

Exercise | Q 163.(c) | Page 271

Supply the missing information for diagram.

Exercise | Q 163.(d) | Page 271

Supply the missing information for diagram.

Exercise | Q 164.(a) | Page 272

If possible, find a hook-up of prime base number machine that will do the same work as the given stretching machine. Do not use (× 1) machines

Exercise | Q 164.(b) | Page 272

If possible, find a hook-up of prime base number machine that will do the same work as the given stretching machine. Do not use (× 1) machines

Exercise | Q 164.(c) | Page 272

If possible, find a hook-up of prime base number machine that will do the same work as the given stretching machine. Do not use (× 1) machines

Exercise | Q 164.(d) | Page 272

If possible, find a hook-up of prime base number machine that will do the same work as the given stretching machine. Do not use (× 1) machines

Exercise | Q 165 | Page 272

Find two repeater machines that will do the same work as a (× 81) machine.

Exercise | Q 166 | Page 272

Find a repeater machine that will do the same work as a (xx 1/8) machine.

Exercise | Q 167 | Page 272

Find three machines that can be replaced with hook-ups of (× 5) machines.

Exercise | Q 168 | Page 272

The left column of the chart lists the lengths of input pieces of ribbon. Stretching machines are listed across the top. The other entries are the outputs for sending the input ribbon from that row through the machine from that column. Copy and complete the chart.

 Input Length Machine × 2 1 5 3 15 14 7
Exercise | Q 169 | Page 273

The left column of the chart lists the lengths of input chains of gold. Repeater machines are listed across the top. The other entries are the outputs you get when you send the input chain from that row through the repeater machine from that column. Copy and complete the chart.

 Input Length Repeater Machine × 23 40 125 2 162
Exercise | Q 170 | Page 273

Long back in ancient times, a farmer saved the life of a king’s daughter. The king decided to reward the farmer with whatever he wished. The farmer, who was a chess champion, made an unusal request:
“I would like you to place 1 rupee on the first square of my chessboard, 2 rupees on the second square, 4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares. Each square should have twice as many rupees as the previous square.” The king thought this to be too less and asked the farmer to think of some better reward, but the farmer didn’t agree.
How much money has the farmer earned?
[Hint: The following table may help you. What is the first square on which the king will place at least Rs 10 lakh?]

 Position of Squareon chessboard Amount(in Rs) 1st square 1 2nd square 2 3rd square 4
Exercise | Q 171 | Page 273

The diameter of the Sun is 1.4 × 109 m and the diameter of the Earth is 1.2756 × 107 m. Compare their diameters by division.

Exercise | Q 172 | Page 273

Mass of Mars is 6.42 × 1029 kg and mass of the Sun is 1.99 × 1030 kg. What is the total mass?

Exercise | Q 173 | Page 274

The distance between the Sun and the Earth is 1.496 × 108 km and distance between the Earth and the Moon is 3.84 × 108 m. During solar eclipse the Moon comes in between the Earth and the Sun. What is distance between the Moon and the Sun at that particular time?

Exercise | Q 174 | Page 274

A particular star is at a distance of about 8.1 × 1013 km from the Earth. Assuring that light travels at 3 × 108 m per second, find how long does light takes from that star to reach the Earth.

Exercise | Q 175 | Page 274

By what number should (–15)–1 be divided so that the quotient may be equal to (–5)–1?

Exercise | Q 176 | Page 274

By what number should (–8)–3 be multiplied so that the product may be equal to (–6)–3?

Exercise | Q 177.(i) | Page 274

Find x.

(- 1/7)^-5 ÷ (- 1/7)^-7 = (-7)^x

Exercise | Q 177.(ii) | Page 274

Find x.

(2/5)^(2x + 6) xx (2/5)^3 = (2/5)^(x + 2)

Exercise | Q 177.(iii) | Page 274

Find x.

2^x + 2^x + 2^x = 192

Exercise | Q 177.(iv) | Page 274

Find x.

(-6/7)^(x - 7) = 1

Exercise | Q 177.(v) | Page 274

Find x.

2^(3x) = 8^(2x + 1)

Exercise | Q 177.(vi) | Page 274

Find x.

5^x + 5^(x - 1) = 750

Exercise | Q 178.(i) | Page 274

If a = – 1, b = 2, then find the value of the following:

ab + ba

Exercise | Q 178.(ii) | Page 274

If a = – 1, b = 2, then find the value of the following:

ab – ba

Exercise | Q 178.(iii) | Page 274

If a = – 1, b = 2, then find the value of the following:

ab × b

Exercise | Q 178.(iv) | Page 274

If a = – 1, b = 2, then find the value of the following:

ab ÷ ba

Exercise | Q 179.(i) | Page 274

Express the following in exponential form:

(-1296)/14641

Exercise | Q 179.(ii) | Page 274

Express the following in exponential form:

(-125)/343

Exercise | Q 179.(iii) | Page 274

Express the following in exponential form:

400/3969

Exercise | Q 179.(iv) | Page 274

Express the following in exponential form:

(-625)/10000

Exercise | Q 180.(i) | Page 274

Simplify:

[(1/2)^2 - (1/4)^3]^-1 xx 2^-3

Exercise | Q 180.(ii) | Page 274

Simplify:

[(4/3)^-2 - (3/4)^2]^((-2))

Exercise | Q 180.(iii) | Page 274

Simplify:

(4/13)^4 xx (13/7)^2 xx (7/4)^3

Exercise | Q 180.(iv) | Page 274

Simplify:

(1/5)^45 xx (1/5)^-60 - (1/5)^(+28) xx (1/5)^-43

Exercise | Q 180.(v) | Page 247

Simplify:

((9)^3 xx 27 xx t^4)/((3)^-2 xx (3)^4 xx t^2)

Exercise | Q 180.(vi) | Page 274

Simplify:

((3^-2)^2 xx (5^2)^-3 xx (t^-3)^2)/((3^-2)^5 xx (5^3)^-2 xx (t^-4)^3

Exercise

## NCERT solutions for Mathematics Exemplar Class 8 chapter 8 - Exponents and Powers

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