56.4 Repeated Radical Elimination

Repeated Radical Elimination is the extension of the single even-root solving technique to equations containing two radical expressions, applying the isolate-and-power procedure twice in sequence, once to remove the first radical and, after the equation is rearranged, a second time to remove whatever radical remains. Because a single application of squaring or cubing only removes a radical that stands completely alone on one side of the equation, a second radical still present after the first application requires its own separate isolation and power step before the equation becomes a pure polynomial equation.

This technique treats the two-radical case as two applications of the same fundamental procedure rather than as an entirely new method, chaining isolation and power-raising steps together until every radical has been cleared.

Handling the First Radical

First Radical Isolation

One of the two radical expressions in the equation is isolated on one side, with every other term, including the second radical, moved to the opposite side, following the same isolation procedure used for a single-radical equation.

First Radical Power Application

Both sides of the equation are raised to the power matching the isolated radical's index, eliminating that first radical, while the second radical, still present on the opposite side, becomes embedded inside the resulting expanded expression.

Handling the Second Radical

Remaining Radical Detection

The equation resulting from the first power application is inspected to confirm whether a radical is still present; if one remains, exactly as expected in the two-radical case, the isolation-and-power procedure must be repeated a second time.

Second Radical Isolation

The remaining radical expression is isolated on its own side of the equation, using the same addition and subtraction techniques as before, with every nonradical term moved to the opposite side.

Second Radical Power Application

Both sides of this newly rearranged equation are raised to the power matching the second radical's index, eliminating it and producing an equation entirely free of radicals.

Finishing the Solve

Final Polynomial Candidate Set

Once every radical has been eliminated, the fully cleared polynomial equation is solved using ordinary linear or quadratic techniques, producing the set of candidate solutions to be screened.

Repeated-Power Candidate Check

Every candidate is substituted back into the original, uneliminated two-radical equation, confirming it produces a true statement; because two separate power-raising steps occurred, checking against the original equation is especially important, since either step independently could have introduced an extraneous solution.