In this explainer, we will learn how to simplify algebraic expressions and solve algebraic equations involving roots, where is a positive integer greater than or equal to 2.
The root is an important mathematical operation that describes the inverse of a power operation with an exponent . Letโs begin by providing a formal definition for the root.
Definition: ๐th Roots
The root of a number , where is a positive integer, is a number that, when raised to the power, gives . We can define this number as , such that
The root is given by .
Note that the root of can equivalently be written as . It is outside the scope of this lesson to express roots using this notation, but if you already are familiar with the notation, it can act as a helpful tool to understanding how the rules of exponents can be applied to expressions involving roots.
Theorem: Properties of ๐th Roots
- When and are well defined and are real numbers, then is also defined such that
- If , then it is also the case that
- When is an odd integer,
- When is an even integer and ,
- When is an even integer and ,
- When is an even integer and ,
In our first two examples, we will demonstrate how to apply a combination of these properties to simplify an expression involving a root.
Example 1: Simplifying Algebraic Expressions Involving Exponents and Cube Roots
Simplify .
Answer
Recall that, for positive real numbers and and positive integers ,
By applying this property in reverse, we can rewrite as
Next, we know that, for odd integers ,
Hence,
Similarly, since ,
Combining these expressions, we obtain
Example 2: Simplifying Algebraic Expressions Involving Exponents and Square Roots
Simplify .
Answer
When a root of the form is given with omitted, we assume . This means we can simplify by applying the property of roots with ,
Thus,
Finally, by writing as and using the property for even , we can simplify as follows:
Hence,
Great care must be taken when finding powers of roots, such as or . In the previous example, we demonstrated that . Since we had an even power inside an even root, the operation was defined for all real values of . If, however, we were simplifying , we would need to perform the following steps:
In this case, inside the square root acts first, ensuring the radicand is positive for all values of . This means we are not taking the even root of a negative number and, in turn, means the resulting function can only output positive values. Since is negative for values of , we must include the absolute value when simplifying, as shown.
In the last two examples, we used a property of roots to express an root as the product of two unique roots. It is important to realize that we can extend this property to express a root as the product of three or more roots in order to simplify an expression. In other words, for positive real numbers , , and and positive integers ,
Example 3: Simplifying Algebraic Expressions with More Than One Variable Involving Exponents and Square Roots
Write in its simplest form.
Answer
Recall that, for positive real numbers , , and and positive integers ,
Since is equivalent to , we can rewrite it as shown:
Next, by using the property for even , we can write
Similarly, can be written as , so
This means . Since , , and the product of absolute value is the absolute value of the products,
In the previous examples, we demonstrated how to use properties of roots to simplify expressions. Letโs now consider an equation involving exponents,
A solution to this equation is found by taking the square root, such that
However, if we substitute into the expression , we obtain . This means that is also a solution to the equation . Therefore, when solving an equation , the solutions include both the positive and negative square roots of . This gives rise to a subtle difference between the two seemingly equivalent statements for integer ,
We can generalize this concept for roots where is an even number.
Theorem: Even and Odd ๐th Roots
Consider the equation for real numbers and and positive integers . The following solutions hold:
| Even | Odd | |
|---|---|---|
| There are no real solutions to the equation. | There is one solution, . | |
| The solutions to the equation are . |
We have seen that we interpret the statements and differently. This gives rise to an additional definition, that of the principal root. This allows us to consider an root as a function by making it, by definition, a one-to-one mapping.
Definition: Principal ๐th Root
Every positive real number has a single positive root, defined . This is known as the principal root.
Letโs demonstrate an application of the properties of even and odd roots in the next example.
Example 4: Simplifying and Solving Equations Involving ๐th Roots
Find the value (or values) of if .
Answer
To solve this equation, we will apply a series of inverse operations. First, we divide both sides by 12:
Next, we recall that, given an equation for real numbers and and positive integers ,
- if is even, the solutions to the equation are ,
- if is odd, there is just one solution, .
Since is odd, there is one solution to this equation, given by .
Hence, .
By combining the properties of roots and this theorem, we are able solve more complicated equations involving exponents. Letโs demonstrate this with one final example.
Example 5: Simplifying and Solving Equations Involving ๐th Roots
Find the value (or values) of given that .
Answer
We will begin by evaluating the right-hand side of this equation. Since 144 and are both square numbers, we can easily find the principal square root of their product, which is
Next, we recall that, given an equation for real numbers and and positive integers , if is even, the solutions to the equation are . Since the expression on the left-hand side of the equation has an even exponent , we will need to take the positive and negative square root of 36.
In particular,
To find the solutions to these two equations, we will next multiply both sides by 5 and then subtract 9:
Hence, the solutions are and .
Letโs finish by recapping some key concepts from this explainer.
Key Points
- The root (or radical) of a number , where is a positive integer, is a number that, when raised to the power, gives . We can define this number as , such that
- The root is given by .
- Every positive real number has a single positive root, defined . This is known as the principal root.
- When and are well defined and are real numbers, is also defined such that
- If , then it is also the case that
- When is an odd integer,
- When is an even integer and ,
- When is an even integer and ,
- When is an even integer and ,
- The solutions to the equation for real
numbers and and positive
integers are as follows:
Even Odd There are no real solutions to the equation. There is one solution,
.The solutions to the equation are .