In week one of math 11 pre-calculus, we were taught to identify, work through, and transform mixed and entire radicals. Entire radicals are square roots with a coefficient of one. For example, if you have the equation (√81), it is an entire radical because its coefficient is no more than one.
Only having a coefficient of one also applies to cubed roots and for any radical with any index. An index is the number three in the equation (³√27), and the radicand is 27. If the index is four in this example, the radical would be a digit multiplied by itself four times. The index does not change whether the equation is mixed or entire because only the coefficient can change that.
More examples of entire radicals are (√106), (³√244), or (√99). These equations do not have to be a perfect square root.
Mixed radicals are equations whose coefficients are a number larger than one. For example, (2√48) and (5√11) are mixed radicals because their coefficient is more than one.
To change an entire radical into a mixed, you must find a perfect square that can multiply into the radicand. If you take the example of the entire radical; (√99), and you want to change it into a mixed radical, you must find the perfect square that can multiply into it. For this equation example, the number nine would be acceptable because nine multiplied by eleven equals ninety-nine. Nine would be its perfect square because the square root of nine is a whole number, this being three.
Taking this information, you would use the number eleven and make it into the radicand for the new mixed radical, (√11). After this, you need the coefficient. Taking the perfect square, nine, from the entire radical example, (√99), and square rooting it to look like this; (√9 = 3).
The three turns into the coefficient for the mixed radical, and the eleven turns into the radicand, creating the equation (3√11). Altogether (3√11) is equal to (√99) in mixed radical form.
If the entire radical number is large and difficult to find a perfect square that multiplies into it, you can create a factor tree to break it down. You can then find any repeating numbers of the prime factors to find a perfect square digit.
Turning a mixed radical into an entire radical is the same process but backward. Take the equation; (4√10), for example. If you want to change it into an entire radical, take the coefficient four and multiply it by itself to find the radical; (√16). Then take the radicand of ten from the equation and multiply it by sixteen. (√16 x √10 = √160). In conclusion, the entire radical for (4√10) is (√160).