lesson
Updated 6 days ago · 2 views
If you double a single penny every day for a month, you end up with over ten million dollars on day 31.
Repeated multiplication grows so fast that writing 2×2×2… becomes unmanageable, which is why we use an exponent (or index) to count the repetitions.
In the expression an, the base a is the factor being multiplied, and the index n counts how many times that factor appears.
📊A clean interactive diagram explaining index anatomy. Shows large text 'aⁿ' with labeled pointers: 'Base (the factor)' pointing to 'a' and 'Index / Exponent (the count)' pointing to 'n'. Below it, shows 'a³ = a × a × a' with 3 distinct color-coded boxes around each 'a'. A small toggle button lets the user switch n from 1 to 5 to see the expansion grow dynamically. Light theme (#f8f9fa background, #1e2945 text, accent #2563eb).
What happens when you multiply or divide terms that already have exponents?
The Three Core Index Laws
According to the Product Law, when multiplying powers with the same base, keep the base and add the exponents: am⋅an=am+n.
The Quotient Law states that when dividing powers with the same base, keep the base and subtract the exponents: anam=am−n (where a=0).
Under the Power Law (or Power of a Power Law), when raising a power to another power, multiply the exponents: (am)n=am⋅n.
📊An interactive 3-card comparison visual of the core exponent rules. Card 1: 'Product Rule' showing x³ · x² = (x·x·x)·(x·x) = x⁵. Card 2: 'Quotient Rule' showing x⁵ / x² = (x·x·x·x·x)/(x·x) = x³. Card 3: 'Power of a Power' showing (x²)³ = (x·x)·(x·x)·(x·x) = x⁶. Clicking each card highlights the matching expanded factors with subtle color grouping (#2563eb for factors, #10b981 for totals). Responsive design, max width 350px.
In 1637, French philosopher and mathematician René Descartes popularized the modern superscript notation an in La Géométrie, replacing tedious repeated letters with the concise rules we use today.
What happens if we subtract exponents and end up with zero, or even a negative number?