nth Root - Simplified Form of A Radical Expression

Simplified Form of A Radical Expression

A radical expression is said to be in simplified form if

  1. There is no factor of the radicand that can be written as a power greater than or equal to the index.
  2. There are no fractions under the radical sign.
  3. There are no radicals in the denominator.

For example, to write the radical expression in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

Next, there is a fraction under the radical sign, which we change as follows:

Finally, we remove the radical from the denominator as follows:

When there is a denominator involving surds it may be possible to find a factor to multiply both numerator and denominator by to simplify the expression. For instance using the factorization of the sum of two cubes:

Simplifying radical expressions involving nested radicals can be quite difficult. It is not immediately obvious for instance that:

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