Power Play (Exponents) | FIO

Question 17

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

Solution

Understand the Question

$1\text{ arab} = 1\text{ billion} = 10^9 = 1{,}000{,}000{,}000\text{ seconds}$ .
To find the date $1\text{ billion seconds}$ ago, we first convert the seconds into years (accounting for leap years with an average of $365.25\text{ days/year}$ ).
We then subtract the calculated duration (years, months, days) from the current reference date.

Step 1 · Convert 1 Billion Seconds to Years

Calculate the number of seconds in one average year ( $365.25\text{ days}$ ):

\begin{aligned} 1\text{ hour} &= 60 \times 60 = 3600\text{ seconds} \\[0.6em] 1\text{ day} &= 24 \times 3600 = 86{,}400\text{ seconds} \\[0.6em] 1\text{ year} &= 365.25 \times 86{,}400 = 31{,}557{,}600\text{ seconds} \end{aligned}

Convert $10^9\text{ seconds}$ to years:

\begin{aligned} \text{Number of years} &= \dfrac{1{,}000{,}000{,}000}{31{,}557{,}600} \\[0.6em] &= 31.69987\dots \approx 31.7\text{ years} \end{aligned}

Step 2 · Calculate the Past Date

Assuming the current date is July 29, 2025, we subtract $31.7\text{ years}$ :

Subtract $31\text{ full years}$ : $2025 - 31 = 1994 \implies \text{July 29, 1994}$
Convert the remaining $0.7\text{ years}$ into months: $0.7 \times 12 = 8.4\text{ months} \approx 8.5\text{ months}$
Go back $8\text{ months}$ from July 29, 1994: $\text{July} - 8\text{ months} = \text{November 29, 1993}$
Go back an additional $0.5\text{ months}$ ( $0.5 \times 30 = 15\text{ days}$ ): $29 - 15 = 14 \implies \text{November 14, 1993}$

Answer

Around mid-November 1993 (approximately $31.7\text{ years}$ ago)

Common Mistakes

Ignoring Leap Years: Using $365\text{ days}$ instead of the average $365.25\text{ days}$ per year leads to an accumulating error over a span of $30+$ years.
Number System Confusion: Misinterpreting $1\text{ arab}$ (Indian system) as $10^{11}$ or $10^{10}$ ; $1\text{ arab} = 10^9 = 1\text{ billion}$ .
Year-Boundary Subtraction: Forgetting that subtracting months going backward past January wraps around into the preceding calendar year.

More questions in FIO

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

If $12^2 = 144$ , what is

(i) $(1.2)^2$

(ii) $(0.12)^2$

(iii) $(0.012)^2$

(iv) $120^2$

Q10

Circle the numbers that are the same—

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

Q11

Identify the greater number in each of the following—

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

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

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

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 $0$ – $9$ , how many digits should the code consist of?

Q13

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?

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

(i) $20^9$

(ii) $10^{11}$

(iii) $10^{10}$

(iv) $10^{18}$

(v) $2 \times 10^9$

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

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?