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

In 2ⁿ, n is known as ______.

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

3^–2 can be written as ______.

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

The value of 3⁵ ÷ 3^–6 is ______.

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

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

The multiplicative inverse of 10^–100 is ______.

The value of (–2)^{2×3 – 1} is ______.

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

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

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

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

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

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

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

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

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

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

(–9)³ ÷ (–9)⁸ is equal to ______.

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

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

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

The standard form for 0.000064 is ______.

The standard form for 234000000 is ______.

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

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

`(3/4)^5 ÷ (5/3)^5` is equal to ______.

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

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

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

Cube of `-1/2` is ______.

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

The multiplicative inverse of 10¹⁰ is ______.

a³ × a^–10 = ______.

5⁰ = ______.

5⁵ × 5^–5 = ______.

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

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

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

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

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

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

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

[2^–1 + 3^–1 + 4^–1]⁰ = ______.

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

The standard form of 12340000 is ______.

The usual form of 3.41 × 10⁶ is ______.

The usual form of 2.39461 × 10⁶ is _______.

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

By multiplying `(5/3)^4` by ______ we get 5⁴.

3⁵ ÷ 3^–6 can be simplified as ______.

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

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

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

The standard form for 0.000000008 is ______.

The usual form for 2.3 × 10^–10 is ______.

On dividing 8⁵ by ______ we get 8.

On multiplying ______ by 2^–5 we get 2⁵.

The value of [3^–1 × 4^–1]² is ______.

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

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

The expression for 3⁵ with a negative exponent is ______.

The value for (–7)⁶ ÷ 7⁶ is ______.

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

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

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

`10^-2 = 1/100`

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

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

(–5)^–2 × (–5)^–3 = (–5)^–6 

(–4)^–4 × (4)^–1 = (4)⁵  

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

5⁰ = 5

(–2)⁰ = 2

`(-8/2)^0 = 0`

(–6)⁰ = –1

(–7)^–4 × (–7)² = (–7)^–2 

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

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

a^p × b^q = (ab)^pq 

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

`a^m = 1/a^-m`

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

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

The standard form for 203000 is 2.03 × 10⁵.

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

The value of 5^–2 is equal to 25.

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

a^m × b^m = (ab)^m

Solve the following:

100^–10 

Solve the following:

2^–2 × 2^–3

Solve the following:

`(1/2)^-2 ÷ (1/2)^-3`

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

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

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

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

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

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

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

(2⁵ ÷ 2⁸) × 2^–7 

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

(2⁵ ÷ 2⁸) × 2^–7

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

Find the value of x so that (–2)³ × (–2)^–6 = (–2)^{2x – 1}

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

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

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

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

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

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

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

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

Write 0.000005678 in the standard form.

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

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

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.

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

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

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

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)

Express the following in standard form:

The mass of a proton in gram is `1673/1000000000000000000000000000`

Express the following in standard form:

A Helium atom has a diameter of 0.000000022 cm.

Express the following in standard form:

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

Express the following in standard form:

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

Express the following in standard form:

Express 56 km in m.

Express the following in standard form:

Express 5 tons in g.

Express the following in standard form:

Express 2 years in seconds.

Express the following in standard form:

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

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

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

Find the value of n.

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

Find the value of n.

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

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

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

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

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?

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?

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?

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?

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.

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.

1. Make a table showing the insect’s location for the first 10 hops.
2. Where will the insect be after n hops?
3. Will the insect ever get to 1? Explain.

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

1. Describe patterns you see in the ones digits of the powers.
2. Predict the ones digit in the following:
   1. 4¹²
   2. 9²⁰
   3. 3¹⁷
   4. 5¹⁰⁰
   5. 10⁵⁰⁰
3. Predict the ones digit in the following:
   1. 31¹⁰
   2. 12¹⁰
   3. 17²¹
   4. 29¹⁰ 

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

1. Write the mass of each planet and the Moon in scientific notation.
2. Order the planets and the moon by mass, from least to greatest.
3. Which planet has about the same mass as earth?

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

1. Complete the table by expressing the distance from each planet to the Sun in scientific notation.
2. Order the planets from closest to the sun to farthest from the sun.

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

1. Which is the heaviest element?
2. Which element is lighter, Silver or Titanium?
3. List all five elements in order from lightest to heaviest.

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

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

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

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

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?

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.

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

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.

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.

Explain why the expression 300 × 2^–n can be used to find the amount of the substance that remains after n half-lives.

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? 

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?

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.

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

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

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

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

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

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

25(5^{n – 2}) = 5ⁿ

Fill in the blanks

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

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.

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

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?

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?

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?

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?

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

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

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

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?

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?

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?

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

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

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

Ajay had a 1 cm 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?

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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.

Supply the missing information for diagram.

Supply the missing information for diagram.

Supply the missing information for diagram.

Supply the missing information for diagram.

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.

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.

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.

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.

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

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

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

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.

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.

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?]

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.

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

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?

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.

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

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

Find x.

`(- 1/7)^-5 ÷ (- 1/7)^-7 = (-7)^x`

Find x.

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

Find x.

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

Find x.

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

Find x.

2^3x = 8^{2x + 1}

Find x.

5^x + 5^{x – 1} = 750

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

a^b + b^a 

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

a^b – b^a 

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

a^b × b^a 

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

a^b ÷ b^a 

Express the following in exponential form:

`(-1296)/14641`

Express the following in exponential form:

`(-125)/343`

Express the following in exponential form:

`400/3969`

Express the following in exponential form:

`(-625)/10000`

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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