lesson
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If you multiply two irrational numbers like 2 and 8, you might expect an endless, messy decimal. Instead, their product is simply 4.
How do two square roots combine to make a clean whole number? The secret lies in a single property that connects multiplication and roots.
The Product Rule for Radicals
The radicand is the expression resting inside the radical sign, while the index is the small number indicating the root degree (like 2 for square roots or 3 for cube roots).
The Product Property of Radicals states that when two radicals share the same index n, you can multiply their radicands under a single root: na⋅nb=na⋅b (where a,b≥0 if n is even).
This property is grounded in rational exponents: since na=a1/n, radical multiplication directly mirrors exponent rules with na⋅nb=a1/n⋅b1/n=(ab)1/n=nab.
📊Interactive diagram
Back in the 14th century, French mathematician Nicole Oresme first conceptualized fractional exponents, uncovering the foundation for why roots multiply just like standard powers.
What happens when numbers sit both outside and inside the radicals? Let's tackle terms with coefficients.
Multiplying Terms with Coefficients
A coefficient is a numerical factor multiplying a variable or radical. When multiplying radical terms, multiply the outside coefficients together, and multiply the inside radicands together.
Let's multiply 36⋅210. First, multiply the outer numbers: 3⋅2=6. Then multiply inside: 6⋅10=60.
Now simplify 60 by factoring out its largest perfect square: 60=4⋅15=215. Combining everything gives 6⋅215=1215.