Power Play (Exponents)

How many times can you fold it over and over?

Estu says “I heard that a sheet of paper can’t be folded more than 7 times”.

Roxie replies “What if we use a thinner paper, like a newspaper or a tissue paper?”

Try it with different types of paper and see what happens.

Think about how many combinations are possible in different contexts. Some examples are—

(i) Pincodes of places in India—The Pincode of Vidisha in Madhya Pradesh is 464001. The Pincode of Zemabawk in Mizoram is 796017.

(ii) Mobile numbers.

(iii) Vehicle registration numbers.

Try to find out how these numbers or codes are allotted/generated.

Tremendous in Ten!

Find a partner to play this game with. In 10 seconds, the person who writes a number or an expression, using only the digits 0-9 and arithmetic operations, that gives a number that is the larger between the two wins the round.

Below are some conditions that you may consider for different rounds.

(i) Exponents are not allowed. Only addition is allowed.

(ii) Exponents are not allowed. Only addition and multiplication are allowed.

(iii) Exponents are allowed. Only addition is allowed.

(iv) Exponents are allowed. Any arithmetic operation is allowed.

You can create your own conditions and/or involve more people to play together.

Express the following in exponential form:

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

(ii) $y \times y$

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

Express each of the following as a product of powers of their prime factors in exponential form:

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Write the numerical value of each of the following:

(i) $2 \times 10^3$

(ii) $7^2 \times 2^3$

(iii) $3 \times 4^4$

(iv) $(-3)^2 \times (-5)^2$

(v) $3^2 \times 10^4$

(vi) $(-2)^5 \times (-10)^6$

Find out the units digit in the value of $2^{224} \div 4^{32}$? [Hint: $4 = 2^2$]

There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would there be after 40 days?

Write the given number as the product of two or more powers in three different ways. The powers can be any integers.

(i) $64^3$ (ii) $192^8$ (iii) $32^{-5}$

Examine each statement below and find out if it is 'Always True', 'Only Sometimes True', or 'Never True'. Explain your reasoning.

(i) Cube numbers are also square numbers.

(ii) Fourth powers are also square numbers.

(iii) The fifth power of a number is divisible by the cube of that number.

(iv) The product of two cube numbers is a cube number.

(v) $q^{46}$ is both a 4th power and a 6th power ($q$ is a prime number).

Simplify and write these in the exponential form.

(i) $10^{-2} \times 10^{-5}$

(ii) $5^7 \div 5^4$

(iii) $9^{-7} \div 9^4$

(iv) $(13^{-2})^{-3}$

(v) $m^5n^{12}(mn)^9$

If $12^2 = 144$ what is

(i) $(1.2)^2$

(ii) $(0.12)^2$

(iii) $(0.012)^2$

(iv) $120^2$

Circle the numbers that are the same—

$2^4 \times 3^6$            $6^4 \times 3^2$            $6^{10}$            $18^2 \times 6^2$            $6^{24}$

Identify the greater number in each of the following—

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

(ii) $2^8$ or $8^2$

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

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0–9, how many digits should the code consist of?

64 is a square number ($8^2$) and a cube number ($4^3$). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible?

The worldwide population of sheep (2024) is about $10^9$, and that of goats is also about the same. What is the total population of sheep and goats?

(ii) $20^9$

(ii) $10^{11}$

(iii) $10^{10}$

(iv) $10^{18}$

(v) $2 \times 10^9$

(vi) $10^9 + 10^9$

Calculate and write the answer in scientific notation:

(i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing.

(ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.

(iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world.

(iv) Total time spent eating in a lifetime in seconds.

What was the date 1 arab/1 billion seconds ago?

Say you can fold a sheet of paper as many times as you wish. What would its thickness be after 30 folds? Make a guess.

Now, what do you think the thickness would be after 30 folds? 45 folds? Make a guess.

Fill the table below.

Notice the change in thickness after two folds. By how much does it increase?

After any 3 folds, the thickness increases 8 times (= 2 × 2 × 2). Check if that is true.

Which expression describes the thickness of a sheet of paper after it is folded 10 times? The initial thickness is represented by the letter-number $v$.

(i) $10v$

(ii) $10 + v$

(iii) $2 \times 10 \times v$

(iv) $2^{10}$

(v) $2^{10}v$

(vi) $10^2v$

What is $(-1)^5$? Is it positive or negative? What about $(-1)^{56}$?

Is $(-2)^4 = 16$? Verify.

What is $0^2$, $0^5$? What is $0^n$?

$3^7$ can also be written as $3^2 \times 3^5$. Can you reason out why?

Use this observation to compute the following:

(i) $2^9$ (ii) $5^7$ (iii) $4^6$

Write the following expressions as a power of a power in at least two different ways:

(i) $8^6$

(ii) $7^{15}$

(iii) $9^{14}$

(iv) $5^8$

Context: In the middle of a beautiful, magical pond lies a bright pink lotus. The number of lotuses doubles every day in this pond. After 30 days, the pond is completely covered with lotuses.

Q. Write the number of lotuses (in exponential form) when the pond was —

(i) fully covered (ii) half covered

Use the observation $m^a \times n^a = (mn)^a$ to compute the value of $2^5 \times 5^5$.

Simplify $\frac{10^4}{5^4}$ and write it in exponential form.

Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different ways can Roxie dress up?

Hint: Try drawing a diagram like the one above.

Context: Estu says, "Next time, I will buy a lock that has 6 slots with the letters A to Z. I feel it is safer."

Q. How many passwords are possible with such a lock?

What is $2^{100} \div 2^{25}$ in powers of 2?

Why can't $n$ be 0?

Can we write $10^3 = \frac{1}{10^{-3}}$?

We had required $a$ and $b$ to be counting numbers. Can $a$ and $b$ be any integers? Will the generalised forms still hold true?

Write equivalent forms of the following.

(i) $2^{-4}$

(ii) $10^{-5}$

(iii) $(-7)^{-2}$

(iv) $(-5)^{-3}$

(v) $10^{-100}$

Simplify and write the answers in exponential form.

(i) $2^{-4} \times 2^7$

(ii) $3^2 \times 3^{-5} \times 3^6$

(iii) $p^3 \times p^{-10}$

(iv) $2^4 \times (-4)^{-2}$

(v) $8^p \times 8^q$

How many times larger than $4^{-2}$ is $4^2$?

Use the power line for 7 to answer the following questions.

Write these numbers in the same way:

(i) 172 (ii) 5642 (iii) 6374

Context: (i) The Sun is located 30,00,00,00,00,00,00,00,00,00,00,000 m from the centre of our Milky Way galaxy. (ii) The number of stars in our galaxy is 1,00,00,00,00,000. (iii) The mass of the Earth is 59,76,00,00,00,00,00,00,00,00,00,00,000 kg.

Q. Write the large-number facts we read just before in this form.

Context: The distance between the Sun and Saturn is $14,33,50,00,00,000\text{ m} = 1.4335 \times 10^{12}\text{ m}$. The distance between Saturn and Uranus is $14,39,00,00,00,000\text{ m} = 1.439 \times 10^{12}\text{ m}$. The distance between the Sun and Earth is $1,49,60,00,00,000\text{ m} = 1.496 \times 10^{11}\text{ m}$.

Q. Can you say which of the three distances is the smallest?

The number line below shows the distance between the Sun and Saturn ($1.4335 \times 10^{12}\text{ m}$). On the number line below, mark the relative position of the Earth. The distance between the Sun and the Earth is $1.496 \times 10^{11}\text{ m}$.

Express the following numbers in standard form.

(i) 59,853

(ii) 65,950

(iii) 34,30,000

(iv) 70,04,00,00,000

Roxie wonders, “Instead of jaggery if we use 1-rupee coins, how many coins are needed to equal my weight?”. How can we find out?

Context: Roxie wonders, "Instead of jaggery if we use 1-rupee coins, how many coins are needed to equal my weight?".

Q. Would the number of coins be in hundreds, thousands, lakhs, crores, or even more? Make an instinctive guess.

Context: Roxie wonders, "Instead of jaggery if we use 1-rupee coins, how many coins are needed to equal my weight?".

Q. Find the answer by making necessary and reasonable assumptions and approximations for the unknowns. Remember, we are not looking for an exact answer but a reasonably close estimate.

How about measuring to find out the weight of a 1-rupee coin?

Context: Estu asks, "What if we use 5-rupee coins or 10-rupee notes instead? How much money could it be?".

Q. Make an instinctive guess first. Then find out (make necessary and reasonable assumptions about the unknown details and find the answers).

Context: Estu says, "When I become an adult, I would like to donate notebooks worth my weight every year". Roxie says, "When I grow up, I would like to do annadāna (offering grains or meals) worth my weight every year".

Q. How many people might benefit from each of these offerings in a year? Again, guess first before finding out.

Context: Roxie and Estu overheard someone saying—"We did padayātra for about 400 km to reach this place! We arrived early this morning."

Q. How long ago would they have started their journey?

Context: Roxie and Estu overheard someone saying—"We did padayātra for about 400 km to reach this place! We arrived early this morning."

Q. Find answers by making necessary assumptions and approximations. Do guess first before calculating to check how close your guess was!

How many times can a person circumnavigate (go around the world) the Earth in their lifetime if they walk non-stop? Consider the distance around the Earth as 40,000 km.

Context: Roxie tells Estu about a science-fiction novel she is reading where they build a ladder to reach the moon, "... I wonder if we actually had a ladder like that, how many steps would it have?".

Q. What do you think? Make an instinctive guess first.

Context: Roxie tells Estu about a science-fiction novel she is reading where they build a ladder to reach the moon, "... I wonder if we actually had a ladder like that, how many steps would it have?".

Q. Would the number of steps be in thousands, lakhs, crores, or even more?

Can you come up with some examples of linear growth and of exponential growth?

With a global human population of about $8 \times 10^9$ and about $4 \times 10^5$ African elephants, can we say that there are nearly 20,000 people for every African elephant?

Calculate and write the answer using scientific notation:

(i) How many ants are there for every human in the world?

(ii) If a flock of starlings contains 10,000 birds, how many flocks could there be in the world?

(iii) If each tree had about $10^4$ leaves, find the total number of leaves on all the trees in the world.

(iv) If you stacked sheets of paper on top of each other, how many would you need to reach the Moon?

A different way to say your age!

"How old are you?" asked Estu. "I completed 13 years a few weeks ago!" said Roxie. "How old are you?" asked Estu again. "I'm 4840 days old today!" said Roxie. "How old are you?" asked Estu again. "I'm ______ hours old!" said Roxie.

Make an estimate before finding this number.

Estu: "I am 4070 days old today. Can you find out my date of birth?"

If you have lived for a million seconds, how old would you be?

$10^5$ seconds $\approx 1.16$ days and $10^6$ seconds $\approx 11.57$ days. Think of some events or phenomena whose time is of the order of: (i) $10^5$ seconds (ii) $10^6$ seconds

Write them in scientific notation.

A fossil of Kelenken Guillermoi, a type of terror bird, is dated to 15 million years ago.

Plants on land started 47 crore/470 million years ago ( $\approx$ _________________ seconds).

Calculate and write the answer using scientific notation:

(i) If one star is counted every second, how long would it take to count all the stars in the universe? Answer in terms of the number of seconds using scientific notation. (ii) If one could drink a glass of water (200 ml) every 10 seconds, how long would it take to finish the entire volume of water on Earth?

Context: Observe the names million ($10^6$), billion ($10^9$), trillion ($10^{12}$), quadrillion ($10^{15}$), quintillion ($10^{18}$), sextillion ($10^{21}$), septillion ($10^{24}$), octillion ($10^{27}$), nonillion ($10^{30}$), decillion ($10^{33}$).

Q. What does the first part of each name denote?