Operations with Radicals – Adding, Subtracting and Multiplying Roots

Radicals can be combined using arithmetic, but only under specific rules. This page closes out the Exponents, Powers and Roots section by bringing everything together: adding and subtracting like radicals works exactly like combining like terms in algebra, while multiplying radicals works by combining the radicands directly underneath a single root.

Adding and Subtracting Like Radicals

Radicals with the same radicand (“like radicals”) combine by adding or subtracting their coefficients, leaving the radical part unchanged: a√x + b√x = (a+b)√x.

Simplify 5√3 + 2√3.

Both terms share √3, so add the coefficients: 5 + 2 = 7. Result: 7√3.

Multiplying Radicals

√a × √b = √(a × b) – multiply the radicands together under a single root.

Simplify √3 × √12.

√3 × √12 = √(3 × 12) = √36 = 6.

Real-Life Application

Geometry: combining multiple radical side-lengths in a perimeter calculation.
Physics: combining radical terms that appear in motion and energy formulas.
Advanced algebra: simplifying expressions before solving equations that involve roots.

Key Takeaways

Only like radicals (same radicand) can be added or subtracted directly.
Multiplying radicals combines the radicands under one root: √a × √b = √(ab).
This completes the Exponents, Powers and Roots section – these skills feed directly into Scientific Notation, covered next.

Practice: Operations with Radicals

Add Like Radicals