Power Play

1. Experiencing the Power Play — folding paper

Take a paper sheet of thickness $0.001$ cm and fold it. Each fold doubles the thickness:

After $17$ folds it is about $131$ cm (taller than you); after $26$ folds about $670$ m (compare the Burj Khalifa at $830$ m); after $30$ folds about $10.7$ km (the height planes fly, and more than the $11$ km depth of the Mariana Trench); after $46$ folds it crosses $7{,}00{,}000$ km. It feels impossible that so few folds reach such enormous sizes — but the thickness grows by multiplying (doubling), not by adding a fixed amount. This is called exponential growth or multiplicative growth. Notice that after every $10$ folds the thickness increases $1024$ times, because $\underbrace{2\times2\times\dots\times2}_{10\text{ times}}=2^{10}=1024$. After $3$ folds it increases $2\times2\times2=8$ times, and this "increase factor" itself is a power of $2$.

2. Exponential notation — base and power

Repeated multiplication of the same number is written compactly:

For example $5^4=5\times5\times5\times5=625$. Here 5 is the base and 4 is the exponent (power); we read $5^4$ as "5 raised to the power 4" or "5 to the power 4" or "the 4th power of 5". The familiar square $n^2$ (read "$n$ squared") and cube $n^3$ (read "$n$ cubed") are just the cases $a=2$ and $a=3$. The fold thickness when folded $7$ times is $0.001\times2^7=0.128$ cm, and folded $10$ times is $0.001\times2^{10}=1.024$ cm.

Watch the sign of the base. A bracket matters:

• $(-4)^3=(-4)\times(-4)\times(-4)=-64$ — odd power of a negative number is negative.
• $(-2)^4=16$ — even power of a negative number is positive.
• $(-1)^{56}=1$ (even), $(-1)^{5}=-1$ (odd).

Different bases can be combined too: $a\times a\times a\times b\times b=a^3b^2$, and $a\times a\times b\times b\times b\times b=a^2b^4$. A handy reminder: $4+4+4=3\times4=12$ (that's multiplication), whereas $4\times4\times4=4^3=64$ (that's a power) — addition and powers are very different!

3. Prime factors in exponential form

Any number can be written as a product of prime powers. Repeatedly divide by the smallest prime:

"The Stones that Shine": $3$ daughters, each with $3$ baskets, each basket $3$ keys, each opening $3$ rooms gives $3^4=81$ rooms, and continuing the chain to diamonds gives $3\times3\times\dots=3^7=2187$ — exponents capture this "3 of everything" pattern instantly.

4. Law 1 — multiplying same base: $n^a\times n^b=n^{a+b}$

When bases are equal, add the exponents. Why? Counting the factors:

So in general:

Example: $3^4\times3^3=3^{7}=81\times27=2187$, and indeed $3^7$ can also be split as $3^2\times3^5$ — any way of breaking $7$ into a sum works.

5. Law 2 — power of a power: $(n^a)^b=n^{ab}$

$4^6$ can be grouped two ways: $(4^3)^2=64^2=4096$ and $(4^2)^3=16^3=4096$. Both equal $4^6$ because $3\times2=2\times3=6$. Generalising:

So $2^{10}=(2^2)^5=(2^5)^2$, and $7^4=(7^2)^2$. A "power of a power" can be reached in more than one way.

6. Law 3 — multiplying same power: $m^a\times n^a=(mn)^a$

From the Magical Pond (doubling pond for 4 days, tripling for 4 more): $2^4\times3^4=(3\times2)^4=6^4$. When the exponents match, multiply the bases:

Example: $2^5\times5^5=(2\times5)^5=10^5=1{,}00{,}000$; and $\dfrac{10^4}{5^4}=\left(\dfrac{10}{5}\right)^4=2^4=16$.

7. Law 4 — dividing same base: $n^a\div n^b=n^{a-b}$

Halving a line of $2^4$ units repeatedly cancels factors of $2$:

So when bases match in a division, subtract the exponents:

Example: $2^{100}\div2^{25}=2^{75}$. Note $n$ cannot be $0$ here, because dividing by $0$ is not allowed.

8. Zero exponent — why $n^0=1$

Apply Law 4 to a number divided by itself: $2^4\div2^4=2^{4-4}=2^0$, but the same is $\dfrac{16}{16}=1$. So:

This works for any non-zero base: $5^0=1$, $(-7)^0=1$, $100^0=1$. (The case $0^0$ is left undefined.)

9. The other side — negative exponents

Continue dividing past the zero power. Halving $2^4$ five times: $2^4\div2^5=2^{4-5}=2^{-1}$, but as a fraction it is $\dfrac{2\times2\times2\times2}{2\times2\times2\times2\times2}=\dfrac12$. So $2^{-1}=\dfrac12$. Likewise $2^4\div2^{10}=2^{-6}=\dfrac{1}{2^6}=\dfrac{1}{64}$. The pattern:

A negative exponent simply means "take the reciprocal". So $10^{-3}=\dfrac{1}{10^3}=\dfrac{1}{1000}$ and $7^{-2}=\dfrac{1}{49}$. On a "power line" for $4$, going down one step divides by $4$: $\dots,4^2=16,\ 4^1=4,\ 4^0=1,\ 4^{-1}=\tfrac14,\ 4^{-2}=\tfrac{1}{16},\dots$ The three laws of exponents keep holding even when $a$ and $b$ are any integers (positive, zero or negative).

10. Powers of 10 and expanded form

Place value is built from powers of $10$. The expanded form of a number can be written using $10^0,10^1,10^2,\dots$

Decimal places use negative powers of $10$:

Useful big powers: a lakh $=10^5$, a million $=10^6$, a crore $=10^7$, an arab $=10^9$, a billion $=10^9$. Powers of $10$ also let us sense the scale of nature: there are about $2\times10^0$ northern white rhinos left, about $4\times10^5$ African elephants, an $8.2\times10^9$ human population, around $3\times10^{12}$ trees, $2\times10^{16}$ ants and an estimated $2\times10^{23}$ stars in the observable universe — each jump of one in the exponent means ten times as many.

11. Scientific notation (standard form)

Huge numbers are easy to miswrite (you may drop a zero). To read them safely, write them as a number between $1$ and $10$ times a power of $10$:

Steps to convert a large number: place the decimal point just after the first digit, then count how many places it moved — that count is the exponent $y$.

• $5900=5.9\times10^3$
• $20800=2.08\times10^4$
• $80{,}00{,}000=8\times10^6$
• Mumbai's $2$ crore population $=2\times10^7$ (the exponent $7$ tells you the size at a glance).
• Mass of Earth $\approx 5.976\times10^{24}$ kg; Sun–Saturn distance $\approx1.4335\times10^{12}$ m; Sun–Earth $\approx1.496\times10^{11}$ m.

The exponent matters most — it fixes "how big" — while the coefficient $x$ gives the fine detail. To compare standard-form numbers, compare exponents first; the smaller exponent is the smaller number (so $1.496\times10^{11}<1.4335\times10^{12}$).

12. Linear vs exponential growth

A ladder to the Moon ($3{,}84{,}400$ km) with $20$ cm steps needs about $1{,}92{,}20{,}00{,}000$ steps (nearly $2$ billion) — added $20$ cm at a time, this is linear growth (additive): each step contributes the same fixed amount. Yet a paper reaches that same distance in just $46$ folds — exponential growth (multiplicative), where each step doubles the previous total. Exponential growth starts slowly but soon races far ahead of linear growth. Everyday examples of exponential growth include doubling bacteria, compound interest and the spreading lotuses of the Magical Pond; examples of linear growth include filling a tank at a steady rate or saving the same amount each month.

13. Worked examples (NCERT-style)

14. Common mistakes to avoid

• Confusing $n\times a$ with $n^a$: $3\times4=12$ but $3^4=81$ — exponent means repeated multiplication, not multiplying base by power.
• Forgetting brackets: $(-2)^4=16$ but $-2^4=-16$. The bracket decides whether the sign is part of the base.
• Writing $n^a\times n^b=n^{a\times b}$ — wrong; you add exponents when multiplying same base.
• Thinking $n^0=0$ — it is $1$ (for $n\neq0$).
• Reading a negative exponent as a negative number: $2^{-3}=\tfrac18$, not $-8$.
• In standard form, leaving the coefficient outside $1\le x<10$ — e.g. writing $59\times10^2$ instead of $5.9\times10^3$.

15. Quick revision checklist

• $n^a$: base $n$, exponent $a$, means $n$ multiplied by itself $a$ times.
• Same base: multiply $\Rightarrow$ add powers; divide $\Rightarrow$ subtract powers.
• Power of a power: $(n^a)^b=n^{ab}$.
• Same power: $m^a\times n^a=(mn)^a$.
• $n^0=1$;   $n^{-a}=\dfrac{1}{n^a}$.
• Standard form: $x\times10^y$ with $1\le x<10$; compare exponents to order numbers.
• Exponential growth is multiplicative and beats linear (additive) growth quickly.