Before we attempt to simplify the given radical expression, let's consider one of the more difficult parts of correctly simplifying radicals. When the exponent of a variable inside the radical is even and the simplified expression for that variable has an odd exponent, we need to use absolute value symbols.
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sqrt(x^2)= |x| & sqrt(x^3)=x sqrt(x)
sqrt(x^4)= x^2 & sqrt(x^6)=|x^3|This is because we do not want the result to be negative — the range of a square root function is all real numbers greater than or equal to 0. Now, consider the given radical expression.
4sqrt(66g^2h^4)
The exponent of g is even and, in the simplified expression, this variable will have an odd exponent. Therefore, when we remove g from the radical, we will need absolute value symbols. The exponent of h is even and, in the simplified expression, it will have an even exponent. Therefore, when we remove h from the radical, we will not need absolute value symbols.
