Radical Expressions?

Question

What difficulties might you with grasping the concepts used in radical expressions. How would you explain square roots, cube roots, n^th roots, and radicals to a student who is having difficulty understanding these concepts? What are some limitations of square root?

Answer 1

The best long term explanation for roots and radicals is to explain them as reciprocal exponents.

√x = x^(1/2)

Then, show them the rules for exponents:
1.) a^b * a^c = a^(b+c)
2.) (a^b)^c = a^(b*c)
3.) a^c + b^c does not equal (a + b)^c

Rule 1 shows why you can't just multiply 2 dissimilar radicals or a whole number and a radical together to get rid of the radical. ex: √2 * 2 = 2√2 or 2^(3/2).

Rule 2 shows why squaring the square root of a number gives you the number itself, and how you mathematically get rid of radicals into something easier to work with. Ex:
2 = √(x + 3)
2^2 = (√(x + 3))^2
4 = x + 3
x = 1

Rule 3 shows why
* radicals can't just be added together normally: √3 + √5 can't be simplified.
* radicals can't just be combined with whole numbers

In the long run, while √3 may be easier to write, it's easier to mathematically manipulate radicals and work with them as long as you remember while you work that radicals are merely reciprocal exponents. They follow all the same rules, so you only need to remember 1 set of rules to work with both.