Solving Radical Expressions

Whether you are working with Algebra 1, 2 or Geometry, at some point during a student's mathematics education, he/she will be asked to solve radical expressions.  The radical expression can be one of square root, cube root, or the nth root, but the process of determining the simplest form of this type of radical expression is the same and should not be feared by the students.

Let us start by working through some examples:

Example 1: sqrt(108). Let's find the factor trees for number 108.

108 = 2 * 54 = 2 * 2 * 27 = 2 * 2 * 3 * 9 = 2 * 2 * 3 * 3 * 3

When we put the number inside square root, the expression becomes

sqrt(108) = sqrt(2 * 2 * 3 * 3 * 3)

Let us group the number in groups of 2 using the same number.  So you will see 1 group of 2, and 1 group of 3, and one remaining 3.  When you are able to group the numbers into groups of 2, you will take one of the numbers out of the group and put it outside the square root and discard the other one. Since we have 1 group of 2, and 1 group of 3, we will take out a 2 and 3, and discard its partners.  When we take out the 2 and 3, we will end up multiplying both of these numbers.   So our example 1 becomes

sqrt(108) = sqrt[(2 * 2) * (3 * 3) * 3] = 2 * 3 sqrt(3) = 6sqrt(3)

Now, let's work on square root of an expression such as

example 2:  sqrt(4 * x^2 * y^3)

Again, we use the same process to break down what is inside the square root by finding the factor tree.

sqrt(4 * x^2 * y^3) = sqrt(2 * 2 * x * x * y *  y * y)

Since we are looking for square root of the expression, we will group the same number/variable in groups of 2.

sqrt(2 * 2 * x * x * y *  y * y) = sqrt[(2 * 2) * (x * x) * (y * y) * y]

Now we will take out one of the values in the groups of 2, and discard the other one.

sqrt[(2 * 2) * (x * x) * (y * y) * y] = 2 * x * y * sqrt(y) = 2xy * sqrt(y)

Now what is the difference is we are looking for the cube root of a number/expression?   When we are evaluating for the square root, we are grouping our factors into groups of 2.  But when we are looking for the cube root, we are grouping the factors into groups of 3.  Once we group the same number/variable in groups of 3, we will take out one of the number/variable outside of the cube root, and then discard the other two.  If we are looking for the nth root of a number/expression, we will still look to break down the original number/expression into its factor tree.  Then we will group the factors in groups of n.  Once we are able to do so, we will take out one of the factors from the group, and discard the other n-1 number/variables of the group.

In order to get more comfortable with this type of problem, I would suggest that the students look for more problems to solve.  In the meantime, you can try these examples on your own:

1. sqrt(125 * x^4 * z)
2. cuberoot(1000 * A^6 * B^4)
3. sqrt(1024 * z^5 * y^2)
4. 5throot(2018 * x^21 * y^10)