## 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 VariableOutside Radical | Exponent of VariableInside Radical | Need Absolute Valueof VariableOutside 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 |

*Simplifying Radicals*

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 VariableOutside the Radical | Exponent of VariableInside the Radical | Need Absolute Value ofVariable Outside the Radical |
---|---|---|---|

x | Odd | Odd | Yes |

y | Even | None | No |

z | Odd | None | Yes |

*Example 1 – Simplifying Radicals*

### 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 VariableOutside the Radical | Exponent of VariableInside the Radical | Need Absolute Value ofVariable Outside the Radical |
---|---|---|---|

x | Even | None | No |

y | Odd | None | Yes |

*Example 3 – Simplifying Radicals*

## 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.