Exponential form

How would you write $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$? It takes a long line and it is easy to lose count. Math­e­mati­cians write it as $2^{10}$: a small num­ber up high tells you how many times to mul­ti­ply. This short way of writ­ing is called expo­nen­tial form. In this les­son you will learn to read and write it, learn the first few pow­ers of the small num­bers, and write a num­ber as a power.

Expo­nen­tial form#

You have met squares and cubes already. The same idea car­ries on for any num­ber of copies.

$$\begin{aligned} a \times a &= a^2 & &\text{read as "a power 2" or "a squared"} \\ a \times a \times a &= a^3 & &\text{read as "a power 3" or "a cubed"} \\ a \times a \times a \times a &= a^4 & &\text{read as "a power 4"} \\ a \times a \times a \times \cdots \ (n \text{ times}) &= a^n & &\text{read as "a power n"} \end{aligned}$$

In $a^n$, the num­ber $a$ is called the base and $n$ is called the expo­nent. The expo­nent is also called the index or the power.

Writ­ing $5 \times 5 \times 5 \times 5$ is the prod­uct form (or expanded form). Writ­ing $5^4$ is the expo­nen­tial form. Both mean the same num­ber.

Two small facts:

• A num­ber to the power $1$ is the num­ber itself: $7^1 = 7$.
• $1$ to any power is $1$, because $1 \times 1 \times 1 \times \cdots = 1$. So $1^{50} = 1$.

Pow­ers to know#

Pow­ers of 2.

$$\begin{aligned} 2^2 &= 2 \times 2 = 4 \\ 2^3 &= 2 \times 2 \times 2 = 8 \\ 2^4 &= 2 \times 2 \times 2 \times 2 = 16 \end{aligned}$$

Each new power is dou­ble the one before: $2^5 = 32$, $2^6 = 64$, $2^7 = 128$, $2^8 = 256$, $2^9 = 512$, $2^{10} = 1024$.

Pow­ers of 3, 4 and 5. A quick trick: $a^4$ is $a^2 \times a^2$.

For exam­ple, $3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81$, and $4^4 = 16 \times 16 = 256$, and $5^4 = 25 \times 25 = 625$.

Squares and cubes of 6 to 9.

Pow­ers of 10. The expo­nent counts the zeros.

$$10^2 = 100 \qquad 10^3 = 1000 \qquad 10^4 = 10{,}000 \qquad 10^5 = 1{,}00{,}000$$

So one lakh is $10^5$ and one crore, $1{,}00{,}00{,}000$, is $10^7$.

Writ­ing a num­ber as a power#

To write a num­ber as a power of a given base, divide by the base again and again until you reach $1$. Count how many times you divided.

Exam­ple 1#

Write $512$ as a power of $2$.

We divided nine times.

$$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9$$

Exam­ple 2#

Express $243$ in pow­ers of $3$.

Divid­ing by $3$: $243$, $81$, $27$, $9$, $3$, $1$. That is five divi­sions.

$$243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5$$

Exam­ple 3#

Express $1024$ in pow­ers of $4$.

Divid­ing by $4$: $1024$, $256$, $64$, $16$, $4$, $1$. That is five divi­sions, so $1024 = 4^5$.

Notice that $1024$ is also $2^{10}$. The same num­ber can be a power of more than one base.

A prod­uct of prime pow­ers#

Any num­ber can be bro­ken into prime fac­tors and then writ­ten with expo­nents.

Exam­ple 4#

Express $432$ as a prod­uct of pow­ers of prime fac­tors.

Divid­ing by $2$: $432$, $216$, $108$, $54$, $27$. That is four 2s. Then $27 = 3 \times 3 \times 3$.

$$432 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^4 \times 3^3$$

Check: $16 \times 27 = 432$.

Which is greater?#

Work out both pow­ers and com­pare.

Exam­ple 5#

Which is greater, $2^{12}$ or $3^8$?

$2^{12} = 2^{10} \times 2 \times 2 = 1024 \times 4 = 4096$.

$3^8 = 3^4 \times 3^4 = 81 \times 81 = 6561$.

So $3^8$ is greater.

A big­ger expo­nent does not always mean a big­ger num­ber. The base mat­ters too.

Remem­ber. In $a^n$, $a$ is the base and $n$ is the expo­nent. $a^n$ means $a$ mul­ti­plied by itself $n$ times.

Com­mon mis­takes#

• Think­ing $2^5$ means $2 \times 5$. It means five 2s mul­ti­plied: $2^5 = 32$.
• Mix­ing up base and expo­nent. $2^5 = 32$, but $5^2 = 25$.
• Count­ing the num­bers instead of the divi­sions when writ­ing a num­ber as a power.
• Think­ing the big­ger expo­nent always wins. $3^8$ is big­ger than $2^{12}$.

Prac­tice#

1. Write $3 \times 3 \times 3 \times 3 \times 3 \times 3$ in expo­nen­tial form.
2. In $7^4$, name the base and the expo­nent.
3. Find the value of $2^7$.
4. Find the value of $5^4$.
5. Express $625$ as a power of $5$.
6. Express $729$ as a power of $3$ and as a power of $9$.
7. Express $4096$ as a power of $4$.
8. Write one lakh as a power of $10$.
9. Express $360$ as a prod­uct of pow­ers of prime fac­tors.
10. Which is greater, $2^5$ or $5^2$?

Answers#