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Rational Exponents and Radicals

Rational Exponents and Radicals 1.8 - Solution

arrow_back Return to Rational Exponents and Radicals

For any real number the radical expression can be simplified as follows. Since the radical is a real number and the index of the root is even, the expression underneath the radical is positive. Otherwise, the radical would be imaginary. With this in mind, let's consider the possible values of the variables, and

  • In the radical, the index is even and the exponent of is even. Therefore, the expression will be real whether the value of is positive, negative or equal to
  • In the radical, the index is even and the exponents of and are odd. Since is always positive, in order for this radical expression to result in a real number, the product of and must be also positive — and must have the same sign.
This means that if we remove and from the radical, we will need absolute value symbols.
Notice that both and have even exponents, so is always positive. Therefore, we don't need absolute value symbols.