# How do I Simplify Radicals? (with pictures)

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.

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