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:

Rational Exponents

Rational (fractional) exponents, indicate that we have a radical expression.

YouTube Playlist on Fractional Exponents

They are intimidating at first but you will quickly get used to working with them. The denominator is the index and the numerator is the exponent.

[ Video Instruction: Rational Exponents ]

Rewrite the radical expression using rational exponents.

Convert to a radical and then simplify.

When simplifying, you can use the rules of exponents to simplify or you can convert to a radical and then simplify. Both methods yield the same result.

Assume all variables are positive and use the rules of exponents to simplify.

Simplify. (Assume all variables are positive.)

Notice that all of the above problems worked out nicely because the exponents were multiples of the root.  This is not always the case as illustrated below.

A quick way to simplify radicals is to divide the index into the exponents. This will tell us what the exponent of the base should be outside the radical and the remainder will be the exponent of the base inside the radical. All the rules of exponents hold true when working with rational exponents.

Sometimes we will be asked to multiply radicals with different indices.  This may seem impossible at first, but it can be done using the rules of exponents as follows.

Video Examples on YouTube: