An exponent is a compact way to write repeated multiplication, and a root simply runs that machine in reverse. Master the pair and big numbers stop being frightening.
Repeated multiplication
When you multiply the same number by itself again and again, writing it all out gets tedious fast. An exponent is the shorthand:
\[ 2^4 = 2 \times 2 \times 2 \times 2 = 16 \]The big number is the base, and the small raised number is the exponent (or power). The exponent counts how many copies of the base are multiplied together. We read \( 2^4 \) as "two to the fourth power", or just "two to the fourth".
Two special cases are worth memorising. A base raised to the first power is just itself, \( b^1 = b \), because there is only one copy. And anything raised to the zero power equals one, \( b^0 = 1 \) for any non-zero base — a result that feels odd now but will make perfect sense once you meet the laws below.
Slide the controls below to change the base and the exponent, and watch the multiplication and its result update in real time.
Squares and cubes
The two smallest exponents are so common they earned their own names, borrowed from geometry. Raising to the power of two is called squaring: \( 5^2 = 25 \) is "five squared", and it is exactly the area of a \( 5 \times 5 \) square — five rows of five.
Raising to the power of three is called cubing: \( 2^3 = 8 \) is "two cubed", and it is the volume of a \( 2 \times 2 \times 2 \) cube — two layers, each two by two. That picture is where the names come from, and it is worth holding onto: an exponent of two builds an area, an exponent of three builds a volume.
Square roots: undoing a square
Every operation in mathematics has a way of being undone, and the square root is what undoes squaring. A square root asks the question: "what number, times itself, gives this?"
\[ \sqrt{25} = 5 \quad\text{because}\quad 5^2 = 25 \]The numbers that come out perfectly even are called perfect squares. The first ten are worth recognising on sight: \( 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \). Their roots are the whole numbers \( 1 \) through \( 10 \).
Most numbers are not perfect squares, and their roots are not whole at all. For example \( \sqrt{2} \approx 1.414 \), and no matter how far you look, it never settles into a neat fraction — it is irrational. That is normal; most roots are like this.
Tip A square root is the question "what times itself gives this?". If the number isn't a perfect square, the answer won't be a whole number — and that is perfectly fine. Recognising the first ten perfect squares makes most root problems instant.
The first laws of exponents
Three rules let you combine powers without writing everything out. They all follow from the same idea: an exponent is just a count of factors, so you can count them together.
Multiplying powers of the same base — add the exponents. When the base matches, \( b^m \times b^n = b^{m+n} \). Count the factors to see why:
\[ 2^2 \times 2^3 = (2 \times 2)(2 \times 2 \times 2) = 2^5 = 32 \]Dividing powers of the same base — subtract the exponents. Dividing cancels factors, so \( b^m \div b^n = b^{m-n} \):
\[ 2^5 \div 2^2 = 2^{5-2} = 2^3 = 8 \]A power raised to a power — multiply the exponents. Each copy gets repeated, so \( (b^m)^n = b^{mn} \):
\[ (2^2)^3 = 2^2 \times 2^2 \times 2^2 = 2^6 = 64 \]Notice how the third rule explains \( b^0 \): \( 2^3 \div 2^3 = 2^{3-3} = 2^0 \), and any number divided by itself is \( 1 \).
Powers of ten
Powers of ten are the friendliest of all, because the exponent simply counts the zeros:
\[ 10^1 = 10, \qquad 10^2 = 100, \qquad 10^3 = 1000 \]This is exactly the place-value idea from Stage 1, written compactly: each step up the powers of ten shifts a digit one place to the left. The same shorthand later powers scientific notation, the tool for writing enormous and tiny numbers without drowning in zeros.
In words The exponent counts the factors — it does not multiply the base. So \( 2^3 = 2 \times 2 \times 2 = 8 \), not \( 2 \times 3 = 6 \). Whenever you feel unsure, write the factors out and count them.
- The exponent \( 4 \) tells you to multiply four copies of the base \( 3 \).
- Multiply two at a time: \( 3 \times 3 = 9 \), then \( 9 \times 3 = 27 \), then \( 27 \times 3 = 81 \).
So \( 3^4 = 3 \times 3 \times 3 \times 3 = \mathbf{81} \).
- Ask the root's question: what number, times itself, gives \( 81 \)?
- Test the whole numbers near it: \( 8^2 = 64 \) is too small, \( 9^2 = 81 \) is exactly right.
Since \( 9^2 = 81 \), we have \( \sqrt{81} = \mathbf{9} \).
Practice
Try each one yourself, then reveal the full solution.
1. Evaluate \( 2^5 \).
The exponent \( 5 \) means five copies of the base \( 2 \) multiplied together.
Build it up step by step: \( 2 \times 2 = 4 \), then \( 8 \), then \( 16 \), then \( 32 \).
So \( 2 \times 2 \times 2 \times 2 \times 2 = \mathbf{32} \).
2. Evaluate \( \sqrt{49} \).
Ask: what number, times itself, gives \( 49 \)?
Since \( 7^2 = 49 \), the square root is that number.
So \( \sqrt{49} = \mathbf{7} \).
3. Evaluate \( 10^4 \).
For powers of ten, the exponent counts the zeros.
An exponent of \( 4 \) means a \( 1 \) followed by four zeros.
So \( 10^4 = \mathbf{10000} \).