Final answer:
To determine if a mixed radical or an entire radical is in simplest form, you factor out common factors, eliminate terms where possible to simplify the expression, and then check if the answer can be simplified further.
Step-by-step explanation:
Factoring can be used to determine whether a mixed radical or an entire radical is in simplest form by factoring out common factors. When you have a radical expression, you want to simplify it by looking for perfect square factors (in the case of square roots) or perfect cube factors (for cube roots), among others, depending on the index of the radical. This process involves factoring the number inside the radical into its prime factors and then grouping the factors based on the index of the radical.
To simplify a radical, you would eliminate terms where possible. For instance, if you have √50, you can factor this into √(25×2). Since 25 is a perfect square, you can take the square root of 25 outside the radical, simplifying the expression to 5√2, which is in simplest form. After simplifying the radical, you should always check the answer to see if it is reasonable and cannot be simplified further.
Remember, the key to simplifying radicals is to factor the number inside completely and then reduce it by pulling out powers of the index. If no further perfect powers of the index remain inside the radical, the expression is then considered to be in simplest form.