Single Variable
Suppose we simplified a radical expression with the results shown below.
The above example assumes that the result is a positive real number. If a problem does not indicate that the result be positive, then you need to assume that we are dealing with both positive and negative real numbers.
If the problem expresses that the result must be a positive number, then the absolute value must be used when simplifying radical expressions with variables. This ensures that the answer is positive. When working with radical expressions this requirement does not apply to any odd root because odd roots exist for negative numbers. (See the table Simplifying Radicals.)
| Case | Index of Radical | Exponent of Variable Outside Radical | Exponent of Variable Inside Radical | Need Absolute Value of Variable Outside Radical |
|---|---|---|---|---|
| 1 | Odd | No | ||
| 2 | Even | Even | No | |
| 3 | Even | Odd | Even | Yes |
| 4 | Even | Odd | Odd | No |
| 5 | Even | Odd | No Variable Left Inside Radical | Yes |
As you can see from the table above, there are only two cases where the absolute value is needed when simplifying a radical: Case 3 and Case 5. Therefore, all we need to look at is if the index of the radical is EVEN, and the exponent of the variable inside the radical is EVEN or there is no variable left inside the radical once the radical is simplified.
NOTE: Any even-numbered root must be a positive number; otherwise, it is imaginary.
Let’s look at a graph of each case. (These can be view on Desmos.)
CASE 1
When the index of the radical is ODD, then no absolute value is needed regardless of the exponent of the variable outside or inside the radical.
CASE 2
When the index of the radical is EVEN and the exponent of the variable outside the radical is EVEN, then no absolute value is needed regardless of the exponent inside the radical.
CASE 3
When the index of the radical and the exponent inside the radical is EVEN, and the exponent of the variable outside the radical is ODD, then an absolute value is needed. (See Figure 3-1.) If the absolute value were not present, then the result of the expression would contain negative values. (See Figure 3-2.)
CASE 4
When the index of the radical is EVEN, and the exponent of the variable outside the radical and the exponent inside the radical is ODD, then no absolute value is needed.
CASE 5
When the index of the radical is EVEN, and the exponent of the variable outside the radical is ODD, and the there is no variable inside the radical, then an absolute value is needed. (See Figure 5-1.) If the absolute value were not present, then the result of the expression would contain negative values. (See Figure 5-2.)
More Than One Variable
If you have more than one variable in your radical expression, then each variable follows the same rules as in the table Simplifying Radicals.
Example 1
With the index of the radical being EVEN
The expression simplifies to
| Variable | Exponent of Variable Outside the Radical | Exponent of Variable Inside the Radical | Need Absolute Value of Variable Outside the Radical |
|---|---|---|---|
| x | Odd | Odd | Yes |
| y | Even | None | No |
| z | Odd | None | Yes |
Example 2
With the index of the radical being ODD
The expression simplifies to
And no absolute value is needed on the variables outside the radical.
Example 3
With the index of the radical being EVEN
The expression simplifies to
| Variable | Exponent of Variable Outside the Radical | Exponent of Variable Inside the Radical | Need Absolute Value of Variable Outside the Radical |
|---|---|---|---|
| x | Even | None | No |
| y | Odd | None | Yes |
Principal Square Root
Every positive number b has two square roots , denoted √b and −√b. The principal square root of b is the positive square root, denoted √b .
The concept of principal square root cannot be extended to real negative numbers since the two square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point +i and -i can then be distinguished. Since either choice is possible, there is no ambiguity in defining i as “the” square root of -1.
References
⭐ “5.2: Simplifying Radical Expressions”. 2017. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Algebra/Book%3A_Advanced_Algebra/05%3A_Radical_Functions_and_Equations/5.02%3A_Simplifying_Radical_Expressions.
“9.6 – Imaginary And Complex Numbers | Hunter College – MATH101”. 2022. courses.lumenlearning.com. https://courses.lumenlearning.com/cuny-hunter-collegealgebra/chapter/16-4-1-complex-numbers/.
“How to Use Absolute Value to Simplify Square Roots of Perfect Square Monomials”. 2022. Study.com. https://study.com/skill/learn/using-absolute-value-to-simplify-square-roots-of-perfect-square-monomials-explanation.html.
“Principal Square Root — From Wolfram MathWorld”. 2022. mathworld.wolfram.com. https://mathworld.wolfram.com/PrincipalSquareRoot.html.
“Principal Value Of A Square Root”. 2022. varsitytutors.com. https://www.varsitytutors.com/hotmath/hotmath_help/topics/principal-value-of-a-square-root.
“Roots and Radicals”. 2022. saylordotorg.github.io. https://saylordotorg.github.io/text_intermediate-algebra/s08-01-roots-and-radicals.html.
“Simplify Radical Expressions | Intermediate Algebra”. 2022. courses.lumenlearning.com. https://courses.lumenlearning.com/intermediatealgebra/chapter/read-or-watch-squares-cubes-and-beyond/.
⭐ “Simplifying Radical Expressions – Examples, Definition, Variables, Steps “. 2022. CUEMATH. https://www.cuemath.com/algebra/simplifying-radical-expressions/.
Videos
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
Mathematical Mysteries 