Class 8 › Mathematics › Chapter 2
Power Play
Exponential growth, exponential notation, the laws of exponents (including zero and negative exponents), powers of 10, scientific notation, and estimating very large quantities.
1. Experiencing the Power Play: Exponential Growth
Fold a sheet of paper again and again. Each fold doubles the thickness. Take the starting thickness as $0.001$ cm.
| Folds | 1 | 5 | 10 | 17 | 20 | 26 | 30 | 40 | 46 |
|---|---|---|---|---|---|---|---|---|---|
| Thickness | 0.002 cm | 0.032 cm | 1.024 cm | ≈ 131 cm | ≈ 10.5 m | ≈ 670 m | ≈ 10.7 km | ≈ 10,995 km | > 7,00,000 km |
- Doubling pattern: after $n$ folds the thickness is $0.001 \times 2^n$ cm.
- After any 3 folds the thickness grows $2 \times 2 \times 2 = 8$ times. After any 10 folds it grows $2^{10} = 1024$ times (e.g. $0.001 \to 1.024$ cm, $1.024$ cm $\to 10.485$ m).
- The big surprise: after 30 folds the paper is as high as a plane flies ($\approx 10.7$ km) and after 46 folds it is thicker than the distance to the Moon ($3{,}84{,}400$ km).
- This is multiplicative (exponential) growth: the quantity is multiplied by the same number at every step.
Real limit: in practice a sheet cannot be folded more than about 7 times. The 46-fold result is a thought experiment, assuming you could fold it as often as you wish.
2. Exponential Notation
Definition
$$n^a = \underbrace{n \times n \times \dots \times n}_{a\text{ times}}$$
Here $n$ is the base and $a$ is the exponent (power). $5^4 = 5 \times 5 \times 5 \times 5 = 625$.
- Reading: $5^4$ is "5 raised to the power 4", "5 to the power 4" or "the 4th power of 5". $n^2$ = $n$ squared, $n^3$ = $n$ cubed.
- Folding: thickness after 10 folds with starting thickness $v$ is $2^{10}v$ (not $10v$ or $10^2v$).
- Different letters: $a \times a \times a \times b \times b = a^3b^2$.
- Power is not multiplication: $4 + 4 + 4 = 3 \times 4 = 12$ but $4 \times 4 \times 4 = 4^3 = 64$.
- Negative bases: $(-4)^3 = -64$. A negative base with an odd exponent gives a negative result; with an even exponent, a positive one. So $(-1)^5 = -1$, $(-1)^{56} = 1$, $(-2)^4 = 16$.
- Powers of 0: $0^2 = 0$, $0^5 = 0$; in general $0^n = 0$ for a counting number $n$.
- Prime factorisation in exponential form: $32400 = 2^4 \times 3^4 \times 5^2$. Also $648 = 2^3 \times 3^4$, $405 = 3^4 \times 5$, $540 = 2^2 \times 3^3 \times 5$, $3600 = 2^4 \times 3^2 \times 5^2$.
"The Stones that Shine": each level multiplies by 3, so diamonds $= 3^7 = 2187$ and rooms $= 3^4 = 81$.
3. Laws of Exponents
Law 1: Same base, multiply
$$n^a \times n^b = n^{a+b}$$
$p^4 \times p^6 = p^{10}$. So $3^7 = 3^4 \times 3^3 = 81 \times 27 = 2187$, and $2^9 = 2^4 \times 2^5 = 16 \times 32 = 512$.
Law 2: Power of a power
$$(n^a)^b = (n^b)^a = n^{a \times b}$$
$4^6 = (4^3)^2 = (4^2)^3 = 4096$. Also $2^{10} = (2^5)^2 = (2^2)^5$. Many numbers can be written as a power of a power in more than one way: $8^6 = (8^2)^3 = (2^3)^6 = 2^{18}$.
Law 3: Same exponent
$$m^a \times n^a = (mn)^a \qquad \frac{m^a}{n^a} = \left(\frac{m}{n}\right)^a$$
$2^5 \times 5^5 = 10^5$ and $\dfrac{10^4}{5^4} = 2^4 = 16$. Regrouping: $3^4 \times 2^4 = (3 \times 2)^4 = 6^4$.
Law 4: Same base, divide
$$n^a \div n^b = n^{a-b} \quad (n \neq 0)$$
$2^4 \div 2^3 = 2^{4-3} = 2$; $\;2^{100} \div 2^{25} = 2^{75}$.
- Why $n \neq 0$? If $n = 0$, dividing gives $\tfrac{0}{0}$, which is not defined.
- These laws hold for all integer exponents (positive, zero and negative), not just counting numbers.
- Reasoning proof: $3^7 = (3 \times 3) \times (3 \times 3 \times 3 \times 3 \times 3) = 3^2 \times 3^5$.
4. Counting Combinations (the Multiplication Principle)
Rule: if one choice can be made in $m$ ways, and for each of them a second choice in $n$ ways, the total number of combinations is $m \times n$. For several independent choices, multiply them all.
- Outfits: 4 dresses and 3 caps give $4 \times 3 = 12$. 7 dresses, 2 hats and 3 pairs of shoes give $7 \times 2 \times 3 = 42$.
- Digit locks (digits 0 to 9, repeats allowed): 2 digits give $10^2 = 100$ codes; 3 digits give $10^3 = 1000$; a 5-digit lock gives $10^5 = 1{,}00{,}000$ passwords (00000 to 99999).
- Letters: a lock with 6 slots of letters A to Z gives $26^6$ codes.
- Alphanumeric code of length 5: $26$ letters $+$ $10$ digits $= 36$ choices per place, so $36^5$ codes.
- How long must a code be? For $8.5$ billion milk packets we need $10^n \geq 8{,}500{,}000{,}000$. Since $10^9 < 8.5 \times 10^9 < 10^{10}$, at least 10 digits are needed.
- Problem-solving tip: if a problem is hard, solve a simpler version first (2-digit lock before 5-digit lock) and look for the pattern.
5. The Other Side of Powers: Zero and Negative Exponents
Why $x^0 = 1$
Use Law 4 on equal powers: $2^4 \div 2^4 = 2^{4-4} = 2^0$.
But anything divided by itself is $1$, so $2^0 = 1$. In general $x^a \div x^a = x^{a-a} = x^0$, hence $x^0 = 1$ for $x \neq 0$.
Why $n^{-a} = \dfrac{1}{n^a}$
A line of length $2^4$ halved 5 times: $2^4 \div 2^5 = \dfrac{2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2 \times 2} = \dfrac{1}{2}$.
By Law 4 it is also $2^{4-5} = 2^{-1}$. So $2^{-1} = \tfrac{1}{2}$. Similarly $2^4 \div 2^{10} = 2^{-6} = \tfrac{1}{2^6} = \tfrac{1}{64}$.
Negative exponents
$$n^{-a} = \frac{1}{n^a} \qquad n^{a} = \frac{1}{n^{-a}} \qquad (n \neq 0)$$
$10^{-3} = \tfrac{1}{10^3}$, $\;7^{-2} = \tfrac{1}{7^2}$, $\;(-7)^{-2} = \tfrac{1}{(-7)^2}$, $\;(-5)^{-3} = \tfrac{1}{(-5)^3}$.
- Power line of 4: $\dots, 4^{-2} = \tfrac{1}{16},\ 4^{-1} = \tfrac{1}{4},\ 4^0 = 1,\ 4^1 = 4,\ 4^2 = 16,\ 4^3 = 64,\ 4^4 = 256,\ 4^5 = 1024,\ 4^6 = 4096,\ 4^7 = 16384,\ 4^8 = 65536$. Going up one step multiplies by the base; going down one step divides by it.
- Using a power line: $4^7 \div 4^5 = 4^2$, so $16384$ is $16$ times $1024$. And $4^2$ is $4^2 \div 4^{-2} = 4^4$ times larger than $4^{-2}$.
- Using the power line of 7: $2401 \times 49 = 7^4 \times 7^2 = 7^6$; $\;49^3 = 7^6$; $\;343 \times 2401 = 7^7$; $\;\dfrac{16807}{49} = 7^3$; $\;\dfrac{7}{343} = 7^{-2}$; $\;\dfrac{1}{343} \times \dfrac{1}{343} = 7^{-6}$.
- Simplifying examples: $2^{-4} \times 2^7 = 2^3$; $\;3^2 \times 3^{-5} \times 3^6 = 3^3$; $\;p^3 \times p^{-10} = p^{-7}$; $\;2^4 \times (-4)^{-2} = 2^4 \times \tfrac{1}{16} = 1$; $\;8^p \times 8^q = 8^{p+q}$; $\;10^{-2} \times 10^{-5} = 10^{-7}$; $\;9^{-7} \div 9^4 = 9^{-11}$; $\;(13^{-2})^{-3} = 13^6$; $\;m^5 n^{12}(mn)^9 = m^{14}n^{21}$.
6. Powers of 10 and Expanded Form
Place values are powers of $10$: ones $= 10^0$, tens $= 10^1$, hundreds $= 10^2$, tenths $= 10^{-1}$, hundredths $= 10^{-2}$, and so on.
Expanded form with powers of 10
$$47561 = 4 \times 10^4 + 7 \times 10^3 + 5 \times 10^2 + 6 \times 10^1 + 1 \times 10^0$$
$$561.903 = 5 \times 10^2 + 6 \times 10^1 + 1 \times 10^0 + 9 \times 10^{-1} + 0 \times 10^{-2} + 3 \times 10^{-3}$$
- Digits after the decimal point use negative powers of 10.
- Examples: $172 = 1 \times 10^2 + 7 \times 10^1 + 2 \times 10^0$; $\;5642 = 5 \times 10^3 + 6 \times 10^2 + 4 \times 10^1 + 2 \times 10^0$.
7. Scientific Notation (Standard Form)
Standard form
$$x \times 10^y, \qquad 1 \leq x < 10,\ y \text{ an integer}$$
$5900 = 5.9 \times 10^3$, $\;20800 = 2.08 \times 10^4$, $\;80{,}00{,}000 = 8 \times 10^6$.
- How to convert: move the decimal point until exactly one non-zero digit is before it. The number of places moved is the exponent (positive for big numbers, negative for numbers less than 1).
- Why use it: long numbers are easy to misread (miscounted zeros or commas). The exponent shows the size at a glance, and the exponent matters more than the coefficient. Changing $2 \times 10^7$ to $3 \times 10^7$ adds half again; changing it to $2 \times 10^8$ makes it 10 times larger.
- Digits show accuracy: $1.42 \times 10^5$ says we know the number to about the thousands; $1.4 \times 10^5$ says we know it less precisely.
- Comparing: compare the exponents first; if they are equal, compare the coefficients. Sun to Earth $= 1.496 \times 10^{11}$ m is far smaller than Sun to Saturn $= 1.4335 \times 10^{12}$ m, and Saturn to Uranus $= 1.439 \times 10^{12}$ m is the largest of the three.
- Examples: $59{,}853 = 5.9853 \times 10^4$; $\;65{,}950 = 6.595 \times 10^4$; $\;34{,}30{,}000 = 3.43 \times 10^6$; $\;70{,}04{,}00{,}00{,}000 = 7.004 \times 10^{10}$.
Calculating in scientific notation
Multiply the coefficients and add the exponents: $(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$.
Example: clothes for everyone: $8.2 \times 10^9 \times 30 = 2.46 \times 10^{11}$. Honeybees: $50{,}000 \times 10^8 = 5.0 \times 10^{12}$. Always rewrite the answer so that $1 \leq x < 10$.
8. Linear Growth vs Exponential Growth
| Linear growth | Exponential growth | |
|---|---|---|
| Change per step | Same amount added each time | Same factor multiplied each time |
| Earth to Moon example | Ladder with steps of $20$ cm: $1{,}92{,}20{,}00{,}000$ steps for $3{,}84{,}400$ km | Paper folded: just $46$ folds |
| Pattern | $20 + 20 + 20 + \dots$ | $0.001 \times 2 \times 2 \times 2 \times \dots$ |
| Chapter examples | Ladder steps | Folding paper, Stones that Shine, Magical Pond, lock combinations |
- Magical Pond: lotuses double daily and cover the pond on day $30$ (that is $2^{30}$ lotuses). On day $29$ the pond is half covered ($2^{29}$). Exponential growth looks slow, then explodes at the end.
- Mixing growth rates: a lotus doubles for 4 days ($2^4$), then triples for 4 days ($\times 3^4$). Total $2^4 \times 3^4 = (2 \times 3)^4 = 6^4 = 1296$. The order of the stages does not matter.
9. Getting a Sense of Large Numbers and Estimation
Estimation steps (for quantities you cannot measure exactly):
1. Guess instinctively (hundreds? lakhs? crores?).
2. Model: write the relationship between the quantities (worth $=$ weight in kg $\times$ cost per kg).
3. Assume sensible values for unknowns (weight 45 kg, jaggery ₹70/kg gives ₹3150).
4. Compute and compare with your guess. Different assumptions give different answers, and that is fine.
1. Guess instinctively (hundreds? lakhs? crores?).
2. Model: write the relationship between the quantities (worth $=$ weight in kg $\times$ cost per kg).
3. Assume sensible values for unknowns (weight 45 kg, jaggery ₹70/kg gives ₹3150).
4. Compute and compare with your guess. Different assumptions give different answers, and that is fine.
| Power of 10 | Example (approximate) |
|---|---|
| $10^0$ | Only 2 northern white rhinos left (mid-2025) |
| $10^3$ | Fewer than 3000 Komodo dragons ($\approx 3 \times 10^3$) |
| $10^5$ | African elephants $\approx 4 \times 10^5$ |
| $10^9$ | Starlings $\approx 1.3 \times 10^9$; humans $\approx 8.2 \times 10^9$ |
| $10^{12}$ | Trees on Earth $\approx 3 \times 10^{12}$ |
| $10^{16}$ | Ants $\approx 2 \times 10^{16}$ |
| $10^{23}$ | Stars in the observable universe $\approx 2 \times 10^{23}$ |
| $10^{25}$ | Drops of water on Earth $\approx 2 \times 10^{25}$ |
- Using the numbers: ants per human $= \dfrac{2 \times 10^{16}}{8 \times 10^9} \approx 2.5 \times 10^6$. Starling flocks of 10,000 birds $= \dfrac{1.3 \times 10^9}{10^4} = 1.3 \times 10^5$. Leaves on all trees (each $10^4$) $= 3 \times 10^{12} \times 10^4 = 3 \times 10^{16}$. Time to count $2 \times 10^{23}$ stars at one per second $= 2 \times 10^{23}$ seconds.
- Times in seconds (1 year $\approx 3.15 \times 10^7$ s):
| Seconds | $10^2$ | $10^4$ | $10^6$ | $10^7$ | $10^9$ | $10^{12}$ | $10^{15}$ | $10^{17}$ |
|---|---|---|---|---|---|---|---|---|
| About | 1.6 min (a cup of tea) | 2.7 hours | 11.57 days | 115.7 days (Mangalyaan to Mars: 298 days) | 31.7 years | 31,700 years (early Homo sapiens) | 3.17 crore years (dinosaurs extinct about 6.6 crore years ago) | 3.17 billion years (Universe is 13.8 billion years old) |
- Remember: $10^6$ seconds is under a fortnight, but $10^9$ seconds is about 31 years. A million seconds is about 12 days.
- Order of magnitude: approximating "2-minute noodles" as of the order of $10^2$ seconds is true whether it takes 120 or 900 seconds.
10. A Pinch of History: Naming Large Numbers
Indian system
Hundred times the last
lakh $10^5$, crore $10^7$, arab $10^9$, kharab $10^{11}$, neel $10^{13}$, padma $10^{15}$, shankh $10^{17}$, maha shankh $10^{19}$.
International system
Thousand times the last
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 $10^{33}$.
Giant numbers
Googol
$\text{googol} = 10^{100}$ (atoms in the universe: $10^{78}$ to $10^{82}$). $\text{Googolplex} = 10^{\text{googol}}$.
- Name pattern: the first part of million, billion, trillion, quadrillion… tells how many factors of $1000$ are multiplied together ($1000^2$, $1000^3$, $1000^4$, …).
- Lalitavistara (Buddhist text, 1st century BCE): number names for odd powers of ten up to $10^{53}$ (tallakshana). Mahaviracharya lists 24 terms (up to $10^{23}$). The Jaina Amalasiddhi names every power up to $10^{96}$. The Pali Kaccayana goes to $10^{140}$ (asankhyeya).
- Record banknotes: Hungary's $10^{21}$ pengő (1946, never issued); Zimbabwe's $10^{14}$ dollar note (2009).
11. Summary and Quick Revision
| Rule | Formula | Example |
|---|---|---|
| Product, same base | $n^a \times n^b = n^{a+b}$ | $2^4 \times 2^5 = 2^9$ |
| Power of a power | $(n^a)^b = n^{ab}$ | $(2^5)^2 = 2^{10}$ |
| Quotient, same base | $n^a \div n^b = n^{a-b}$ ($n \neq 0$) | $5^7 \div 5^4 = 5^3$ |
| Product, same exponent | $n^a \times m^a = (nm)^a$ | $2^5 \times 5^5 = 10^5$ |
| Quotient, same exponent | $n^a \div m^a = (n \div m)^a$ ($m \neq 0$) | $10^4 \div 5^4 = 2^4$ |
| Zero exponent | $n^0 = 1$ ($n \neq 0$) | $7^0 = 1$ |
| Negative exponent | $n^{-a} = \dfrac{1}{n^a}$ | $10^{-3} = \dfrac{1}{1000}$ |
| Scientific notation | $x \times 10^y$, $1 \leq x < 10$ | $308100000 = 3.081 \times 10^8$ |
Compare powers
Rewrite, then compare
$3^4 = 81 > 4^3 = 64$. $2^8 = 256 > 8^2 = 64$. $2^{100} > 100^2$ because $2^{100} = 1024^{10}$ is far bigger than $10^4$.
Same number?
Break into primes
$6^4 \times 3^2 = 2^4 \times 3^6$ and $18^2 \times 6^2 = 2^4 \times 3^6$, so these are equal; $6^{10}$ and $6^{24}$ are different.
Units digit
Reduce first
$2^{224} \div 4^{32} = 2^{224} \div 2^{64} = 2^{160}$. Powers of 2 end in $2, 4, 8, 6$ repeating, and $160$ is a multiple of $4$, so the units digit is $6$.
Square and cube
Sixth powers
A number that is both a square and a cube is a 6th power ($n^6$): $1, 64, 729, 4096, \dots$ (infinitely many).
Always / sometimes / never true:
• Cube numbers are square numbers: sometimes (only when they are 6th powers, e.g. $64$, but not $8$).
• Fourth powers are square numbers: always ($n^4 = (n^2)^2$).
• $n^5$ is divisible by $n^3$: always ($n^5 \div n^3 = n^2$).
• The product of two cube numbers is a cube: always ($n_1^3 n_2^3 = (n_1 n_2)^3$).
• $q^{46}$ (q prime) is both a 4th and a 6th power: never, because $46$ is divisible by neither $4$ nor $6$.
• Cube numbers are square numbers: sometimes (only when they are 6th powers, e.g. $64$, but not $8$).
• Fourth powers are square numbers: always ($n^4 = (n^2)^2$).
• $n^5$ is divisible by $n^3$: always ($n^5 \div n^3 = n^2$).
• The product of two cube numbers is a cube: always ($n_1^3 n_2^3 = (n_1 n_2)^3$).
• $q^{46}$ (q prime) is both a 4th and a 6th power: never, because $46$ is divisible by neither $4$ nor $6$.
Common mistakes:
• Writing $a^m + a^n = a^{m+n}$. The law applies to multiplication, not addition. ($10^9 + 10^9 = 2 \times 10^9$, not $10^{18}$ or $20^9$.)
• Confusing $n^3$ with $3n$: $4^3 = 64$, not $12$.
• Mixing up $(-3)^2 = 9$ with $-3^2 = -9$ (the exponent applies only to what is directly before it).
• Multiplying exponents in $a^m \times a^n$ (you must add) or adding them in $(a^m)^n$ (you must multiply).
• Combining different bases and exponents: $2^3 \times 3^2$ is not $6^5$.
• Thinking $a^0 = 0$. It is $1$ (for $a \neq 0$).
• Treating $2^{-3}$ as negative: it equals $\tfrac{1}{8}$, which is positive.
• Leaving scientific notation with a coefficient outside $1 \leq x < 10$ (e.g. $32 \times 10^5$ should be $3.2 \times 10^6$).
• Writing $a^m + a^n = a^{m+n}$. The law applies to multiplication, not addition. ($10^9 + 10^9 = 2 \times 10^9$, not $10^{18}$ or $20^9$.)
• Confusing $n^3$ with $3n$: $4^3 = 64$, not $12$.
• Mixing up $(-3)^2 = 9$ with $-3^2 = -9$ (the exponent applies only to what is directly before it).
• Multiplying exponents in $a^m \times a^n$ (you must add) or adding them in $(a^m)^n$ (you must multiply).
• Combining different bases and exponents: $2^3 \times 3^2$ is not $6^5$.
• Thinking $a^0 = 0$. It is $1$ (for $a \neq 0$).
• Treating $2^{-3}$ as negative: it equals $\tfrac{1}{8}$, which is positive.
• Leaving scientific notation with a coefficient outside $1 \leq x < 10$ (e.g. $32 \times 10^5$ should be $3.2 \times 10^6$).