All About Numbers
Exponents – Repeated Multiplication Made Simple
An exponent is a small raised number that tells you how many times to multiply a number by itself. Instead of writing 5 × 5 × 5 × 5, you can write it far more compactly as 5⁴ (“5 to the power of 4”). What starts as a simple shorthand quickly becomes essential once numbers get large – the difference between writing out twenty 2's multiplied together and simply writing 2²⁰ is the difference between a page of arithmetic and a few keystrokes.
The compact superscript notation we use today (x², x³, and so on) was popularised by the French mathematician and philosopher René Descartes in his 1637 work La Géométrie. Before Descartes, mathematicians had no standard shorthand at all – they wrote out expressions like “the cube of x” in full words, or repeated the letter itself (writing “xx” for x², and even “xxx” for x³), which became hopelessly clumsy for anything beyond small powers. Descartes's clean superscript notation caught on quickly across Europe and has barely changed since.
What Is an Exponent?
In an expression like 5⁴, the exponent (4) tells you how many times the base (5) is multiplied by itself: 5⁴ = 5 × 5 × 5 × 5.
Exponents as Area and Volume
The two smallest exponents have a direct geometric meaning: squaring a number (power 2) gives the area of a square, and cubing a number (power 3) gives the volume of a cube.
Beyond power 3, there's no everyday shape to draw (a "4-dimensional cube" isn't something you can sketch on paper), but the pattern of repeated multiplication continues exactly the same way: n⁴ = n × n × n × n, and so on for any exponent. The Square Numbers and Cube Numbers pages explore the area and volume cases in full detail.
3⁵ = 3 × 3 × 3 × 3 × 3 = 243.
Real-Life Application
- Computing: memory sizes are measured in powers of 2 (1 kilobyte = 2¹⁰ bytes).
- Science: extremely large or small measurements use exponents (light-years, atomic sizes).
- Population growth: repeated doubling is naturally described with exponents.
Key Takeaways
- An exponent tells you how many times to multiply the base by itself.
- The compact x², x³ notation was popularised by Descartes in 1637.
- Exponents turn long repeated multiplication into short, manageable notation.