# 10.2 Solving Exponential Equations

Exponential equations are often reduced by using radicals—similar to using exponents to solve for radical equations. There is one caveat, though: while odd index roots can be solved for either negative or positive values, even-powered roots can only be taken for even values, but have both positive and negative solutions. This is shown below:

Example 10.2.1

Solve for in the equation .

The solution for this requires that you take the fifth root of both sides.

When taking a positive root, there will be two solutions. For example:

Example 10.2.2

Solve for in the equation .

The solution for this requires that the fourth root of both sides is taken.

The answer is because and .

When encountering more complicated problems that require radical solutions,  work the problem so that there is a single power to reduce as the starting point of the solution. This strategy makes for an easier solution.

Example 10.2.3

Solve for in the equation .

The first step should be to isolate , which is done by dividing both sides by 2. This results in .

Once isolated,  take the square root of both sides of this equation:

Checking these solutions in the original equation indicates that both work.

Example 10.2.4

Solve for in the equation .

First, isolate by subtracting 6 from both sides. This results in .

Now,  take the cube root of both sides, which leaves:

Checking this solution in the original equation indicates that it is a valid solution.

Since you are solving for an odd root, there is only one solution to the cube root of −125. It is only even-powered roots that have both a positive and a negative solution.

Solve.