54.4 Like Radical Combination

Like Radical Combination is the technique for adding or subtracting radical expressions, requiring every radical term to first be simplified and then combined only with other terms that share both the identical index and the identical radicand, treating each such matching radical the way a variable term is treated in ordinary polynomial addition. It mirrors like-term combination for polynomials directly, with the radical portion of a term playing the same role a variable's letter-and-exponent portion plays in that earlier context.

Because two radicals only qualify as like radicals when their index and their fully simplified radicand match exactly, this technique places simplification as a mandatory first step, since an unsimplified radical can obscure a match, or a mismatch, that would otherwise be obvious.

Preparing the Terms

Preliminary Radical Simplification

Every radical term in the expression is fully simplified first, extracting any perfect-power factors from each radicand, since two terms that look different before simplification may become identical afterward, or two terms that look similar may turn out to differ once simplified.

Defining a Match

Matching Radical Index

Two radical terms can only be combined if they share the identical index; a square root and a cube root, for example, can never be combined through addition or subtraction regardless of what their radicands happen to be.

Matching Simplified Radicand

Beyond sharing an index, two radical terms can only be combined if their fully simplified radicands are identical; radicals with the same index but different radicands, even ones that seem numerically close, remain unlike terms.

Combining Like Radicals

Like Radical Coefficient Addition

When two like radical terms are added, their exterior coefficients are added together using ordinary arithmetic, and the shared radical portion, index and radicand, is carried forward unchanged.

Like Radical Coefficient Subtraction

When one like radical term is subtracted from another, their exterior coefficients are subtracted the same way, again carrying the shared radical portion forward unchanged into the result.

Handling Terms That Do Not Match

Unlike Radical Term Retention

Any radical term that does not share both the index and the simplified radicand of another term in the expression is left as its own separate term in the final result, exactly as an unlike variable term is left unchanged in polynomial addition.

Presenting the Final Result

Radical Sum-or-Difference Form

The final result of a like-radical combination lists every distinct like-radical group as a single combined term, alongside every unlike radical term left uncombined, presented together as a sum or difference exactly as any simplified polynomial expression would be.