Unit 1: Powers

A shortcut for multiplying

Writing 3 × 3 × 3 × 3 is tiring, and easy to get wrong. So we write 3^4 instead.

The big number is the base — the thing being multiplied. The little one is the exponent — how many of them there are.

So 3^4 means four 3s multiplied together. It does not mean 3 × 4.

Squares and cubes

Two exponents get their own names, because they draw pictures.

5^2 is said five squared — it is the number of dots in a 5 by 5 square.

5^3 is five cubed — the dots in a 5 by 5 by 5 box. That is where the names come from.

Multiplying powers: just count them

2^3 × 2^4 is three 2s multiplied, then four more 2s. Altogether that is seven 2s.

So 2^3 × 2^4 = 2^7.

You did not memorise a rule there. You counted. When the bases match, add the exponents — and now you know why.

Dividing takes them away again

2^5 ÷ 2^2 is five 2s on the top and two 2s underneath. Two of them cancel out, and three are left.

So dividing subtracts the exponents.

And a power of a power multiplies them: (2^3)^4 is four copies of "three 2s", which is twelve 2s.

Why anything to the power 0 is 1

Here is a question that sounds like a trick: what is 7^0?

Use the dividing rule. 7^3 ÷ 7^3 subtracts the exponents and gives 7^0. But any number divided by itself is 1.

So 7^0 = 1. Not because someone decided it — because nothing else would fit.

Going below zero

Keep subtracting and the exponent goes negative. 2^2 ÷ 2^5 leaves three 2s underneath, so 2^-3 means 1/8.

A negative exponent flips the number over. It never makes it negative.

The exponent counts the zeros

10^2 = 100. 10^3 = 1000. 10^6 = 1000000.

With a base of ten, the exponent is simply how many zeros follow the 1. Nothing else behaves so tidily, which is why our whole number system is built on ten.

It is also why a computer scientist prefers powers of two and gets a much less tidy table.

Going the other way makes tiny numbers

A negative exponent flips the number over, so with ten it makes it small.

10^-1 is 1/10, which is 0.1. 10^-3 is 0.001.

Now the exponent counts the places after the point. One rule, running in both directions from 10^0 = 1.

Standard form: a number and a power of ten

Scientists write huge and tiny numbers as one digit, a point, and a power of ten.

The distance to the sun is about 1.5 × 10^8 km. A virus is about 1 × 10^-7 m across.

Both fit on one line, and you can compare them at a glance by looking only at the exponents.

What could a half power mean?

The rules say x^(1/2) × x^(1/2) = x^1. So a half power is the thing that gives x when multiplied by itself.

That is exactly what a square root is. So x^(1/2) = sqrt(x).

Nobody chose that. The rule you already had left no other option.

Top and bottom do different jobs

In x^(2/3) the bottom takes the root and the top raises the power.

So 8^(2/3) means "cube root of 8, then squared": 2^2 = 4.

Do the root first — the numbers stay small and friendly. 8^2 = 64 then a cube root is the same answer by a harder path.

A power spreads over a product

(2 × 5)^3 is (2 × 5)(2 × 5)(2 × 5). Multiplication does not care about order, so gather the 2s and the 5s:

2^3 × 5^3

So (ab)^n = a^n b^n. The power reaches every factor inside the bracket.

The one place it does not work

A power spreads over multiplying. It does not spread over adding.

(3 + 4)^2 is 7^2 = 49. But 3^2 + 4^2 is 9 + 16 = 25.

Not close. This is the single most common mistake in all of algebra, and now you have checked it with numbers rather than been warned about it.

One fact, two ways round

2^5 = 32 answers "what do I get?"

A logarithm asks the other question: "what power was it?" We write that log(2, 32) = 5, and say log base 2 of 32 is 5.

They are the same fact. One says the power, the other asks for it.

Logs turn multiplying into adding

Multiplying powers adds their exponents. A log *is* an exponent. So logs add too:

log(2, 8) + log(2, 4) = log(2, 32), because 8 × 4 = 32.

That is the whole reason logarithms were invented. Before calculators, adding was hours faster than multiplying.

Every exponent rule has a logarithm twin

A logarithm is an exponent, so every rule from lesson 2 comes back wearing a different hat.

Multiplying powers adds exponents, so:

log(b, xy) = log(b, x) + log(b, y)

Dividing subtracts. You do not have to learn these twice — you already know them.

A power comes down to the front

log(b, x^n) is n copies of log(b, x) added together, which is just n × log(b, x).

So an exponent inside a logarithm slides out to the front:

log(2, 8^5) = 5 × log(2, 8) = 5 × 3 = 15

This is the rule that made logarithm tables worth printing: it turns a power into a multiplication.

Logs of numbers below one are negative

log(2, 1/8) asks: what power of 2 gives one eighth?

A negative one. 2^-3 = 1/8, so the answer is -3.

So the log of anything between 0 and 1 is negative, the log of 1 is 0, and above 1 it is positive. There is no log of a negative number at all — no power of 2 will ever give you -4.