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?

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

Practice: Exponents