In order to discuss the simplification of radicals, some important terms must be employed. “Radical” is the term we use to refer to the symbol that denotes a square root or “nth” root, and “radicand” is the number inside the radical symbol. A radical is simplified when the radicand has no remaining square root or nth root factors. In order to simplify radicals, the radicand must be factored, and any factor that is a square root or nth root must be reduced and placed in front of the radical sign. For the purposes of this discussion, square roots will be considered.
When a radicand is a perfect square, it is relatively easy to simplify. The square is reduced, and the radical symbol is removed. When the radicand is not a perfect square, the radicand must be factored in order to determine whether any of the factors can be simplified. Any factors that are a perfect square must be simplified and placed in front of the radical symbol. Factors that are not a perfect square will remain beneath the radical symbol.
For example, 7 is the square root of 49. When a radical is presented with a radicand of 49, simplification involves the removal of the radical sign and the replacement of 49 with 7. Sometimes, however, a radical is presented with a radicand that is not a perfect square. In such cases, it might appear impossible to simplify, but factoring of the radicand can prove that simplification is possible.
A radicand that can be factored can be simplified if any of the factors are a perfect square. A radical with a radicand of 54, for example, can be factored into 9 x 6. In order to show the process of simplification, this equation would appear beneath the radical symbol. Once factored into 9 x 6, the perfect square — 9 — can be moved out from beneath the radical symbol and reduced to result in the integer 3. The 3 would then be placed in front of the radical symbol, and 6 would remain underneath the radical symbol — which you would read as “3 times the square root of 6.”
When attempting to simplify radicals, you might come across a radical that cannot be simplified. For example, a radical with a radicand of 33 cannot be simplified, because 33 has no square factors. Thirty-three can be factored as 3 x 11, but because neither 3 nor 11 is a perfect square, no portion of the radicand can be removed from beneath the radical symbol.