Power Play (Exponents) | FIO

Question 16

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.

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Solution
Understand the Question
  • Scientific Notation: Expresses a number in the form a×10na \times 10^n, where 1≤a<101 \le a < 10 and nn is an integer.
  • For calculations involving the world population, we take the standard value of approximately 8 billion=8×1098 \text{ billion} = 8 \times 10^9 people.
  • Unit conversions:
    • 1 million=1061 \text{ million} = 10^6
    • 1 trillion=10121 \text{ trillion} = 10^{12}

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

Step 1 · Calculate Total Pieces of Clothing

Given:

  • World population≈8×109\text{World population} \approx 8 \times 10^9
  • Clothes per person=30\text{Clothes per person} = 30Diagram 1
Total clothes=(8×109)×30=(8×30)×109=240×109=(2.4×102)×109=2.4×102+9=2.4×1011\begin{aligned} \text{Total clothes} &= (8 \times 10^9) \times 30 \\[0.6em] &= (8 \times 30) \times 10^9 \\[0.6em] &= 240 \times 10^9 \\[0.6em] &= (2.4 \times 10^2) \times 10^9 \\[0.6em] &= 2.4 \times 10^{2+9} \\[0.6em] &= 2.4 \times 10^{11} \end{aligned}
Answer

(i) 2.4×1011 pieces of clothing2.4 \times 10^{11} \text{ 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.

Step 1 · Calculate Total Number of Honeybees

Given:

  • Number of colonies=100 million=100×106=108\text{Number of colonies} = 100 \text{ million} = 100 \times 10^6 = 10^8
  • Bees per colony=50,000=5×104\text{Bees per colony} = 50{,}000 = 5 \times 10^4Diagram 2
Total bees=(Number of colonies)×(Bees per colony)=(1×108)×(5×104)=(1×5)×(108×104)=5×108+4=5×1012\begin{aligned} \text{Total bees} &= (\text{Number of colonies}) \times (\text{Bees per colony}) \\[0.6em] &= (1 \times 10^8) \times (5 \times 10^4) \\[0.6em] &= (1 \times 5) \times (10^8 \times 10^4) \\[0.6em] &= 5 \times 10^{8+4} \\[0.6em] &= 5 \times 10^{12} \end{aligned}
Answer

(ii) 5×1012 bees5 \times 10^{12} \text{ bees}

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

Step 1 · Calculate Total Bacterial Population

Given:

  • World population≈8×109\text{World population} \approx 8 \times 10^9
  • Bacterial cells per human=38 trillion=38×1012=3.8×1013\text{Bacterial cells per human} = 38 \text{ trillion} = 38 \times 10^{12} = 3.8 \times 10^{13}Diagram 3
Total bacteria=(World population)×(Bacterial cells per human)=(8×109)×(3.8×1013)=(8×3.8)×(109×1013)=30.4×109+13=30.4×1022=(3.04×101)×1022=3.04×101+22=3.04×1023\begin{aligned} \text{Total bacteria} &= (\text{World population}) \times (\text{Bacterial cells per human}) \\[0.6em] &= (8 \times 10^9) \times (3.8 \times 10^{13}) \\[0.6em] &= (8 \times 3.8) \times (10^9 \times 10^{13}) \\[0.6em] &= 30.4 \times 10^{9+13} \\[0.6em] &= 30.4 \times 10^{22} \\[0.6em] &= (3.04 \times 10^1) \times 10^{22} \\[0.6em] &= 3.04 \times 10^{1+22} \\[0.6em] &= 3.04 \times 10^{23} \end{aligned}
Answer

(iii) 3.04×1023 bacterial cells3.04 \times 10^{23} \text{ bacterial cells}

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

Step 1 · Convert Daily Eating Time to Seconds

Assuming an average eating time of 1.5 hours/day1.5 \text{ hours/day}:

Eating time per day in seconds=1.5×60×60=1.5×3600=5400 seconds\begin{aligned} \text{Eating time per day in seconds} &= 1.5 \times 60 \times 60 \\[0.6em] &= 1.5 \times 3600 \\[0.6em] &= 5400 \text{ seconds} \end{aligned}

Step 2 · Calculate Total Lifetime Eating Time

Assuming an average lifespan of 70 years70 \text{ years} (365 days/year365 \text{ days/year}):Diagram 4

Number of days in lifespan=70×365=25550 days\begin{aligned} \text{Number of days in lifespan} &= 70 \times 365 \\[0.6em] &= 25550 \text{ days} \end{aligned} Total eating time in lifespan=5400×25550=137,970,000 seconds=1.3797×108 seconds\begin{aligned} \text{Total eating time in lifespan} &= 5400 \times 25550 \\[0.6em] &= 137{,}970{,}000 \text{ seconds} \\[0.6em] &= 1.3797 \times 10^8 \text{ seconds} \end{aligned}
Answer

(iv) 1.3797×108 seconds1.3797 \times 10^8 \text{ seconds}

Common Mistakes
  • Coefficient Range: Forgetting that in scientific notation a×10na \times 10^n, the coefficient aa must satisfy 1≤a<101 \le a < 10 (e.g., leaving 240×109240 \times 10^9 instead of 2.4×10112.4 \times 10^{11}, or 30.4×102230.4 \times 10^{22} instead of 3.04×10233.04 \times 10^{23}).
  • Unit Confusion: Misinterpreting million as 10510^5 instead of 10610^6, or trillion as 10910^9 (billion) instead of 101210^{12}.

More questions in FIO

Q1

Express the following in exponential form:

(i) 6×6×6×66 \times 6 \times 6 \times 6

(ii) y×yy \times y

(iii) b×b×b×bb \times b \times b \times b

(iv) 5×5×7×7×75 \times 5 \times 7 \times 7 \times 7

(v) 2×2×a×a2 \times 2 \times a \times a

(vi) a×a×a×c×c×c×c×da \times a \times a \times c \times c \times c \times c \times d

Q2

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

Q3

Write the numerical value of each of the following:

(i) 2×1032 \times 10^3

(ii) 72×237^2 \times 2^3

(iii) 3×443 \times 4^4

(iv) (−3)2×(−5)2(-3)^2 \times (-5)^2

(v) 32×1043^2 \times 10^4

(vi) (−2)5×(−10)6(-2)^5 \times (-10)^6

Q4

Find out the units digit in the value of 2224÷4322^{224} \div 4^{32}? [Hint: 4=224 = 2^2]

Q5

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

Q6

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

(i) 64364^3 (ii) 1928192^8 (iii) 32−532^{-5}

Q7

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) q46q^{46} is both a 4th power and a 6th power (qq is a prime number).

Q8

Simplify and write these in the exponential form.

(i) 10−2×10−510^{-2} \times 10^{-5}

(ii) 57÷545^7 \div 5^4

(iii) 9−7÷949^{-7} \div 9^4

(iv) (13−2)−3(13^{-2})^{-3}

(v) m5n12(mn)9m^5 n^{12} (mn)^9

Q9

If 122=14412^2 = 144, what is

(i) (1.2)2(1.2)^2

(ii) (0.12)2(0.12)^2

(iii) (0.012)2(0.012)^2

(iv) 1202120^2

Q10

Circle the numbers that are the same—

24×3664×32610182×626242^4 \times 3^6 \qquad 6^4 \times 3^2 \qquad 6^{10} \qquad 18^2 \times 6^2 \qquad 6^{24}

Q11

Identify the greater number in each of the following—

(i) 434^3 or 343^4

(ii) 282^8 or 828^2

(iii) 1002100^2 or 21002^{100}

Q12

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 00–99, how many digits should the code consist of?

Q13

64 is a square number (828^2) and a cube number (434^3). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Q14

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?

Q15

The worldwide population of sheep (2024) is about 10910^9, and that of goats is also about the same. What is the total population of sheep and goats?

(i) 20920^9

(ii) 101110^{11}

(iii) 101010^{10}

(iv) 101810^{18}

(v) 2×1092 \times 10^9

(vi) 109+10910^9 + 10^9

Q16

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.

Q17

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

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