Solving Radical Equations

Now that we have learned how to work with radical expressions, we next move on to solving.

Playlist on Radical Expressions and Equations

Use caution when solving radical equations because the following steps may lead to extraneous solutions, solutions that do not solve the original equation.

Solve: 

Step 1: Isolate the radical.

Step 2: Square both sides of the equation.

Step 3: Solve the resulting equation and then check your answers.

Whenever you raise both sides of an equation to an even power, you introduce the possibility of extraneous solutions so the check is essential here.
  

Solve.

The index determines the power to which we raise both sides.  For example, if we have a cube root we will raise both sides to the 3rd power. The property that we are using is

for integers n > 1 and positive real numbers x. After eliminating the radical, we will most likely be left with either a linear or a quadratic equation to solve.

The check mark indicates that we have actually checked that the value is a solution to the equation, do not dismiss this step, it is essential.

Some radical equations have more than one radical expression.  These require us to isolate each remaining radical expression and raise both sides to the nth power until they are all eliminated.  Be patient with these, go slow and avoid short cuts.
Solve: 

Solve: 

Solve: 

Video Examples on YouTube:

Adding and Subtracting Radical Expressions

To add or subtract radical expressions, simplify first and then add like terms if there are any.  The radical parts of the terms must be exactly the same before we can add them. (Assume all variables are positive.)

Perform the operations.

Be sure to follow the correct order of operations.
  

Simplify.

Unsimplified rational expressions may look as if they have no like terms, but first try simplifying and then check for like terms.

Simplifying is a challenge for many students. In particular, the numerical part is the source of confusion.  It is much easier to deal with the prime factors of the number in a radical than it is with the number itself. Create a factor tree and determine the prime factorization of all numbers before simplifying radical expressions.

Simplify: 

Begin by determining the prime factorizations of 108 and 864.

Substitute the prime factorization in and then simplify.

Simplify.

Video Examples on YouTube: