Meet exponents

Counting $1, 2, 4, 8, 16, 32, \dots$ each step doubles the previous one. After ten doublings you reach $1024$; after twenty, more than a million; after thirty, a billion. Writing all those products out becomes silly. We invented exponents so we can write them once.

Concept

An expression like $7^4$ has two parts:

• The base is the number being multiplied. Here the base is $7$.
• The exponent (or power or index) is how many times the base appears as a factor. Here the exponent is $4$.

So $7^4 = 7 \times 7 \times 7 \times 7 = 2401$. Read it as "seven raised to the power $4$", or simply "seven to the fourth".

Two special powers have their own names:

• $a^2$ is the square of $a$, because it counts the unit squares in an $a \times a$ grid.
• $a^3$ is the cube of $a$, because it counts the unit cubes in an $a \times a \times a$ box.

A few simple facts to internalise:

1. $a^1 = a$. Any number to the first power is itself.
2. $1^n = 1$ for every $n$. Multiplying $1$ by itself never changes anything.
3. $(-1)^n$ alternates: $(-1)^{\text{even}} = 1$, $(-1)^{\text{odd}} = -1$.
4. $0^n = 0$ for any positive $n$. But $0^0$ is left undefined.

When the base is a fraction, the exponent applies to the whole fraction:

$\left(\frac{2}{3}\right)^3 = \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{2^3}{3^3} = \frac{8}{27}.$

When the base is negative, parentheses matter:

$(-2)^4 = (-2)(-2)(-2)(-2) = 16, \qquad -2^4 = -(2^4) = -16.$

The first squares the negative; the second squares the positive then negates.

Powers of $10$ deserve special attention because we use base $10$ for writing numbers. $10^n$ is just "$1$ followed by $n$ zeros":

$10^1 = 10, \quad 10^2 = 100, \quad 10^3 = 1000, \quad 10^6 = 1{,}000{,}000.$

This makes powers of $10$ the perfect tool for compressing very large numbers.

Worked examples

Example 1. Express $32$ as a power of $2$.

• $2 \times 2 \times 2 \times 2 \times 2 = 32$, five factors.
• So $32 = 2^5$.

Example 2. Compute $\left(\frac{3}{5}\right)^4$.

• $\left(\frac{3}{5}\right)^4 = \frac{3^4}{5^4} = \frac{81}{625}$.

Example 3. Find the value of $(-3)^5$ and $-3^5$.

• $(-3)^5 = -3 \times -3 \times -3 \times -3 \times -3$. Five negatives multiply to negative: $-243$.
• $-3^5 = -(3^5) = -243$. Same answer here because $5$ is odd.

Example 4. Express $1728$ as a product of prime powers.

• Prime factorise: $1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^6 \times 3^3$.
• This compact form is much easier to work with than the long product.

Try it yourself

1. Find the value of $5^4$ and $2^7$.
2. Express $625$ as a power of $5$.
3. Compute $\left(\frac{1}{4}\right)^3$.
4. Find $(-2)^6$ and $-2^6$. Are they equal?
5. Write $9000$ as a product of prime powers.
6. Express $0.001$ as a power of $10$.
7. A bacterium divides into two every hour. Starting with one, how many are there after $8$ hours? Express your answer as a power of $2$.
8. A piece of paper $0.1$ mm thick is folded in half $10$ times. How thick is the stack? (Each fold doubles the thickness.)

Activity

Paper folding. Take an A4 sheet and try to fold it in half as many times as possible. After each fold count the layers: $2, 4, 8, 16, \dots$. You will find you cannot fold beyond about $6$ or $7$ times , even though mathematically the layer count is just $2^n$. Discuss with a friend why exponents grow so quickly that even paper "runs out".