Multiplying Radicals
Simple Explanation
To multiply two radicals with the same index, multiply the radicands together under one radical: ⁿ√a · ⁿ√b = ⁿ√(ab). Multiply any coefficients out front separately, then simplify the result if possible.
Why Do We Need It?
Multiplying radicals comes up whenever you combine two measurements or quantities that were each expressed as a root — areas, volumes, and physics formulas frequently need this step.
Formula
Product Rule for Radicals
ⁿ√a · ⁿ√b = ⁿ√(ab)
The product of two radicals with the same index equals the radical of the product of their radicands.
- a, b
- — the radicands (a, b ≥ 0 when n is even)
- n
- — the common index (root) of both radicals
When to use it: Whenever you simplify a radical by factoring out a perfect nth-power factor, or multiply two radicals with the same index.
Worked Example
Multiply two radicals and simplify
Simplify √6 · √15.
Why Does This Work?
The product rule for radicals, ⁿ√a · ⁿ√b = ⁿ√(ab), holds because both sides, raised to the nth power, give exactly ab — so they must be equal (for nonnegative radicands).
Real-Life Example
Finding the area of a rectangle with radical side lengths
A rectangular garden plot has sides √6 m and √15 m.
Its area is √6 × √15 = √90 = 3√10 m² — multiplying radicals directly, then simplifying, gives a clean final answer.
Practice
Simplify √3 · √12.
EasyCommon mistake
Multiplying the radicands but forgetting to also multiply any coefficients out front — (2√3)(4√5) = 8√15, not 2·4 left un-multiplied or dropped entirely.
Quick Review
- ⁿ√a · ⁿ√b = ⁿ√(ab) — multiply the radicands under one radical.
- Multiply coefficients separately: (c·ⁿ√a)(d·ⁿ√b) = cd·ⁿ√(ab).
- Always simplify the resulting radical afterward if possible.