Expressions and Equations
Exponents and powers
Work out powers and use the laws of exponents to simplify them.
A power is a short way of writing the same number multiplied by itself over and over. 2⁵ means 2 × 2 × 2 × 2 × 2. The big number at the bottom is the base — what is being multiplied. The small number up top is the index, and it counts how many of them there are.
The three laws
When the base is the same: multiplying adds the indices, dividing subtracts them, and raising a power to another power multiplies them. They all come from counting how many copies of the base you end up with.
Worked example
Why does 2³ × 2⁴ equal 2⁷?
Write both out in full: (2 × 2 × 2) × (2 × 2 × 2 × 2).
Nothing clever yet — just undoing the shorthand.
Try it together
Let us simplify (3² × 3⁴) ÷ 3³ together.
Every base here is 3, so all three laws are available. Take the brackets first.
1.3² × 3⁴ is 3 to the power of what? Type just the index.
Two odd-looking cases
Any base to the power nought is 1, and a negative index means one over that power, so 2⁻² is 1/4. Both fall straight out of the subtraction law: 2³ ÷ 2³ is 2⁰, and it is also 1.
Have a go
Have a go on your own: what is 5³?
Hint: 5 × 5 = 25, then 25 × 5.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.