In the first week of Pre-Calculus 11, we first reviewed concepts taught in previous years, such as whole numbers, rational numbers, and irrational numbers. We also learned how to convert between entire and mixed radicals. A radical is any expression containing the root symbol √ . An entire radical number refers to one that is completely under the radical sign (e.g. √12, ∛28, and ∜352). A mixed radical is written as a product of a rational number and a radical in the form of an√b. For example, √12 can be converted to a mixed radical as follows:
As seen in the above illustration, the factor tree can be used to identify the factors of √12, which is a product of √4 and √3. Since √4 is a perfect square, √12 can also be expressed as 2√3.
Mixed radicals are not limited to square roots. In our assignment on page 27, we also resolved cube roots, fourth roots, and fifth roots. In Question 8b, ∛56 is converted to a mixed radical as follows:
The operation is undoubtedly increasingly challenging as n in an√b increases. For example, in question 8h:
In this example, although 2 x 2 makes 4, 2 is not a perfect ∜ of 4, therefore ∜4 is left as a radical. In other words, the number of occurrences of a single factor must match the power number of the root.
Note that the occurrences of a single factor can be from different branches of a factor tree:
In this exercise, I learned that every mixed radical can be expressed as an entire radical. Also, if the rational number of a mixed radical is 1, the coefficient needs not be explicitly written.
