Comparing and using powers

Once you can read powers fluently, a whole new world opens up: you can answer questions like "how much bigger is the Sun than the Earth?" or "how many atoms are in this grain of sand?" without writing a single zero.

Concept

Comparing two pure powers. Three little tricks cover almost every case.

1. Same base, compare exponents. $2^{15}$ vs $2^{10}$: same base $2$, so $2^{15}$ is bigger.
2. Same exponent, compare bases. $5^7$ vs $3^7$: same exponent, so $5^7 > 3^7$.
3. Rewrite to share a base or an exponent. $4^{10}$ vs $2^{19}$: rewrite $4^{10} = (2^2)^{10} = 2^{20}$, so $4^{10} > 2^{19}$.

A useful approximation: $2^{10} = 1024 \approx 10^3$. So $2^{50} = (2^{10})^5 \approx (10^3)^5 = 10^{15}$.

Estimating in standard form. When numbers are in $k \times 10^n$ form, compare the $n$'s first; tie-break with the $k$'s. So $4.1 \times 10^9 < 1.2 \times 10^{10}$ even though $4.1 > 1.2$, because $10$ beats $9$.

Growth that uses powers. Several real situations have exponential growth or decay, meaning quantities multiplied (not added) at each step.

• A bacterium that doubles every $20$ minutes: $1, 2, 4, 8, \dots = 2^t$ after $t$ steps.
• Compound interest at $r$ per period: principal $P$ becomes $P(1 + r)^t$ after $t$ periods.
• Radioactive decay: half remains after every "half-life", so $\text{quantity} = q_0 \cdot (1/2)^t$.

The most striking thing about exponential growth is how slow it looks at first and how fast it explodes later. Doubling $1$ rupee daily for a month gives over $1$ billion rupees on day $30$ , but only $\mathbb{?}\,$1024 by day $11$.

Counting tiny things. Many "how many" questions in chemistry and biology are easiest in scientific notation.

• $1$ gram of water contains about $3.3 \times 10^{22}$ molecules.
• Earth has about $1.4 \times 10^9$ km$^3$ of water , far too many to write out without exponents.

Worked examples

Example 1. Which is larger: $3^{10}$ or $10^3$?

• $3^{10}$. Note $3^4 = 81 > 10^2 = 100$? Actually $81 < 100$. But $3^5 = 243 > 10^2$, so $3^{10} = (3^5)^2 = 243^2 = 59{,}049$, much bigger than $10^3 = 1000$.

Example 2. Compare $4^9$ and $2^{20}$.

• $4^9 = (2^2)^9 = 2^{18}$.
• $2^{18}$ vs $2^{20}$: same base, smaller exponent. So $4^9 < 2^{20}$.

Example 3. A culture starts with $200$ bacteria and doubles every hour. How many after $6$ hours?

• After $t$ hours: $200 \cdot 2^t$.
• After $6$: $200 \cdot 64 = 12{,}800$.

Example 4. A $\mathbb{?}\,10{,}000$ deposit earns $10\%$ per year, compounded yearly. How much after $5$ years?

• Amount $= 10{,}000 \cdot (1.1)^5$.
• $(1.1)^5 = 1.61051$, so the amount is $\mathbb{?}\,16{,}105.10$.

Try it yourself

1. Compare $5^{6}$ and $6^{5}$.
2. Compare $2^{30}$ and $10^{9}$ using $2^{10} \approx 10^{3}$.
3. Which is bigger: $9^{8}$ or $3^{17}$?
4. A culture of $100$ bacteria triples every hour. How many after $4$ hours?
5. The number of grains of rice on a chessboard if you put $1$ on square $1$, $2$ on square $2$, $4$ on square $3$, doubling each square. How many on square $32$? Express as a power of $2$.
6. A radioactive sample halves every $10$ years. What fraction remains after $50$ years?
7. Distance from Earth to Pluto $\approx 5.9 \times 10^{9}$ km. At $3 \times 10^{5}$ km/s, how long would light take to reach Pluto (in seconds)?
8. Estimate how many seconds you have lived if your age is $13$ years. (Use $1$ year $\approx 3.15 \times 10^{7}$ s.)

Activity / Insight

Population doubling demo. Place $1$ rupee in a piggy bank today, $2$ tomorrow, $4$ the day after, and so on for $20$ days. Track the running total on a graph. Around day $10$ it still looks small. By day $15$ it is shooting up. By day $20$ you would need over $\mathbb{?}\,10$ lakh in total. This is exponential growth , the engine behind everything from internet users to the spread of a virus.