Power Play (Exponents) | IT

Question 3

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Solution

IT-3

Chapter: POWER PLAY (EXPONENTS)
Class: 8 (Class 8)
Category: in_text

Question

Fill the table below.

Question diagram(s):

Question diagram

The thickness of the paper approximately doubles with each fold.

Step 1 — Identify the pattern

We are given the thickness for several folds. Let $T_n$ be the thickness for Fold $n$ . The thickness for Fold 18 is $\approx 262 \text{ cm}$ . The thickness for Fold 19 is $\approx 524 \text{ cm}$ . We observe that $524 \text{ cm}$ is $2 \times 262 \text{ cm}$ . The thickness for Fold 20 is $\approx 10.4 \text{ m}$ . Let us convert $10.4 \text{ m}$ to centimeters for comparison. $10.4 \text{ m} = 10.4 \times 100 \text{ cm}$ $= 1040 \text{ cm}$ If we double the thickness of Fold 19, we get $2 \times 524 \text{ cm} = 1048 \text{ cm}$ . The given value for Fold 20 ( $1040 \text{ cm}$ ) is approximately $1048 \text{ cm}$ . This confirms that the thickness approximately doubles with each fold. We will use the given approximate values for subsequent calculations, rounding to one decimal place for meters and kilometers.

Step 2 — Calculate thickness for Folds 21 to 26

We start with the thickness for Fold 20, which is $\approx 10.4 \text{ m}$ . $T_{21} \approx 2 \times T_{20}$ $= 2 \times 10.4 \text{ m}$

$\boxed{\approx 20.8 \text{ m}}$

$T_{22} \approx 2 \times T_{21}$ $= 2 \times 20.8 \text{ m}$

$\boxed{\approx 41.6 \text{ m}}$

$T_{23} \approx 2 \times T_{22}$ $= 2 \times 41.6 \text{ m}$

$\boxed{\approx 83.2 \text{ m}}$

$T_{24} \approx 2 \times T_{23}$ $= 2 \times 83.2 \text{ m}$

$\boxed{\approx 166.4 \text{ m}}$

$T_{25} \approx 2 \times T_{24}$ $= 2 \times 166.4 \text{ m}$

$\boxed{\approx 332.8 \text{ m}}$

$T_{26} \approx 2 \times T_{25}$ $= 2 \times 332.8 \text{ m}$

$\boxed{\approx 665.6 \text{ m}}$

Step 3 — Calculate thickness for Folds 28 to 30

We are given the thickness for Fold 27, which is $\approx 1.3 \text{ km}$ . Let us verify this value with our previous calculation. $T_{27} \approx 2 \times T_{26} = 2 \times 665.6 \text{ m} = 1331.2 \text{ m}$ . We convert $1331.2 \text{ m}$ to kilometers. $1331.2 \text{ m} = 1331.2 \div 1000 \text{ km}$ $= 1.3312 \text{ km}$ Rounding $1.3312 \text{ km}$ to one decimal place gives $1.3 \text{ km}$ . This matches the given value. So, we will use $1.3 \text{ km}$ for Fold 27 for the next calculations.

$T_{28} \approx 2 \times T_{27}$ $= 2 \times 1.3 \text{ km}$

$\boxed{\approx 2.6 \text{ km}}$

$T_{29} \approx 2 \times T_{28}$ $= 2 \times 2.6 \text{ km}$

$\boxed{\approx 5.2 \text{ km}}$

$T_{30} \approx 2 \times T_{29}$ $= 2 \times 5.2 \text{ km}$

$\boxed{\approx 10.4 \text{ km}}$

Step 4 — Calculate thickness for Folds 31 to 35

We continue doubling the previous thickness. $T_{31} \approx 2 \times T_{30}$ $= 2 \times 10.4 \text{ km}$

$\boxed{\approx 20.8 \text{ km}}$

$T_{32} \approx 2 \times T_{31}$ $= 2 \times 20.8 \text{ km}$

$\boxed{\approx 41.6 \text{ km}}$

$T_{33} \approx 2 \times T_{32}$ $= 2 \times 41.6 \text{ km}$

$\boxed{\approx 83.2 \text{ km}}$

$T_{34} \approx 2 \times T_{33}$ $= 2 \times 83.2 \text{ km}$

$\boxed{\approx 166.4 \text{ km}}$

$T_{35} \approx 2 \times T_{34}$ $= 2 \times 166.4 \text{ km}$

$\boxed{\approx 332.8 \text{ km}}$

Step 5 — Calculate thickness for Folds 36 to 40

We continue doubling the previous thickness. $T_{36} \approx 2 \times T_{35}$ $= 2 \times 332.8 \text{ km}$

$\boxed{\approx 665.6 \text{ km}}$

$T_{37} \approx 2 \times T_{36}$ $= 2 \times 665.6 \text{ km}$

$\boxed{\approx 1331.2 \text{ km}}$

$T_{38} \approx 2 \times T_{37}$ $= 2 \times 1331.2 \text{ km}$

$\boxed{\approx 2662.4 \text{ km}}$

$T_{39} \approx 2 \times T_{38}$ $= 2 \times 2662.4 \text{ km}$

$\boxed{\approx 5324.8 \text{ km}}$

$T_{40} \approx 2 \times T_{39}$ $= 2 \times 5324.8 \text{ km}$

$\boxed{\approx 10649.6 \text{ km}}$

Step 6 — Calculate thickness for Folds 41 to 45

We continue doubling the previous thickness. $T_{41} \approx 2 \times T_{40}$ $= 2 \times 10649.6 \text{ km}$

$\boxed{\approx 21299.2 \text{ km}}$

$T_{42} \approx 2 \times T_{41}$ $= 2 \times 21299.2 \text{ km}$

$\boxed{\approx 42598.4 \text{ km}}$

$T_{43} \approx 2 \times T_{42}$ $= 2 \times 42598.4 \text{ km}$

$\boxed{\approx 85196.8 \text{ km}}$

$T_{44} \approx 2 \times T_{43}$ $= 2 \times 85196.8 \text{ km}$

$\boxed{\approx 170393.6 \text{ km}}$

$T_{45} \approx 2 \times T_{44}$ $= 2 \times 170393.6 \text{ km}$

$\boxed{\approx 340787.2 \text{ km}}$

Answer

Fold 21: $\approx 20.8 \text{ m}$ Fold 22: $\approx 41.6 \text{ m}$ Fold 23: $\approx 83.2 \text{ m}$ Fold 24: $\approx 166.4 \text{ m}$ Fold 25: $\approx 332.8 \text{ m}$ Fold 26: $\approx 665.6 \text{ m}$ Fold 28: $\approx 2.6 \text{ km}$ Fold 29: $\approx 5.2 \text{ km}$ Fold 30: $\approx 10.4 \text{ km}$ Fold 31: $\approx 20.8 \text{ km}$ Fold 32: $\approx 41.6 \text{ km}$ Fold 33: $\approx 83.2 \text{ km}$ Fold 34: $\approx 166.4 \text{ km}$ Fold 35: $\approx 332.8 \text{ km}$ Fold 36: $\approx 665.6 \text{ km}$ Fold 37: $\approx 1331.2 \text{ km}$ Fold 38: $\approx 2662.4 \text{ km}$ Fold 39: $\approx 5324.8 \text{ km}$ Fold 40: $\approx 10649.6 \text{ km}$ Fold 41: $\approx 21299.2 \text{ km}$ Fold 42: $\approx 42598.4 \text{ km}$ Fold 43: $\approx 85196.8 \text{ km}$ Fold 44: $\approx 170393.6 \text{ km}$ Fold 45: $\approx 340787.2 \text{ km}$

More questions in IT

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 $0^n$ ?

Q10

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

Q11

Use this observation to compute the following:

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

Q12

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(i) $8^6$

(ii) $7^{15}$

(iii) $9^{14}$

(iv) $5^8$

Q13

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.

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(i) fully covered (ii) half covered

Q14

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

Q15

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

Q16

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Hint: Try drawing a diagram like the one above.

Q17

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?

Q18

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

Q19

Why can't $n$ be 0?

Q20

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

Q21

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

Q22

Write equivalent forms of the following.

(i) $2^{-4}$

(ii) $10^{-5}$

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

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

(v) $10^{-100}$

Q23

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) $8^p \times 8^q$

Q24

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

Q25

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

Q26

Write these numbers in the same way:

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

Q27

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.

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Q28

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 $1,49,60,00,00,000\text{ m} = 1.496 \times 10^{11}\text{ m}$ .

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Q29

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Q30

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

Q31

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Q32

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

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Q33

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?

Q34

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

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Q35

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Q36

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

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Q37

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!

Q38

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.

Q39

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?".

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Q40

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Q41

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Q42

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Q43

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

Q44

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"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?"

Q45

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

Q46

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Write them in scientific notation.

Q47

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Q48

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Q49

Calculate and write the answer using scientific notation:

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Q50

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