This document provides information on radicals and root expressions. It defines the key parts of a radical expression, including the index and radicand. It explains that roots and radicals are inverse operations of exponents, and how to "undo" a power or radical. Examples are given of various nth roots and their relationships to exponents. The document outlines the rules for principal roots, noting even roots have two solutions while odd roots only have one. It cautions the reader to check the index when evaluating roots. Finally, it discusses how to simplify nth roots involving variables by dividing the exponent by the index.

1. UNIT 5 - RADICALS
ROOTS AND RADICAL EXPRESSIONS

2. IMPORTANT INFO
 Parts of a Radical Expression:
 Note: If no index is written, it is assumed to be 2
 Example: Identify the index & radicand:
5
87

3. IMPORTANT INFO
 Roots/radicals are the inverse (opposite) operation of applying
exponents/powers.
 You can “undo” a power with a radical; you can “undo” a radical with a power
 Example: 22
= 4 𝑎𝑛𝑑 4 = 22 = 2
 The square root of a number “a” is a number “y” such that 𝑦2 = 𝑎. Cube roots,
4th roots, etc. are also treated similarly

4. IF 𝑎 𝑛
= 𝑏, THEN 𝑎 IS THE NTH ROOT OF 𝑏
 Examples:
 22
= 4, 2 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 4
 23 = 8, 2 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 8
 24 = 16, 2 𝑖𝑠 𝑡ℎ𝑒 4𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 16
 25
= 32, 2 𝑖𝑠 𝑡ℎ𝑒 5𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 32

5. PRINCIPAL ROOTS
 Example: The square root of 25 is written as 25. This has two possible roots
{−5,5}.
 Why?
 The principal root is the positive root.
 The 4th root of 16
4
16 = ±2 because 16 = +2 4 = −2 4. Again, the
principal root is the positive root, 2.
 However, the cube root of 27 (
3
27) only has one root of 3.

6. PRINCIPAL ROOTS - RULES
 Even Roots ( 𝑥, 4
𝑥, 6
𝑥):
 Two solutions (roots)
 Solutions are conjugates (±)
 Your calculator will only show the principal root
 Odd Roots (3
𝑥, 5
𝑥 , 4
𝑥):
 One solution
 Will be positive or negative; based on sign of radicand

7. PRINCIPAL ROOTS - RULES
 WHY?
 A positive or negative number multiplied by itself an even number of times yields a
positive number.
 A positive number multiplied by itself an odd number of times yields a positive number; a
negative number multiplied by itself an odd number of times yields a negative number.

8. CAUTION!
 ALWAYS be sure to check your index to ensure you are taking the correct
root!!
 If we are only simplifying a radical, the principal root is generally accepted
 When we want to solve a radical equation, we must use the ± with even roots

9. EXPONENT TABLE
 Complete the exponent table on the last page of your packet using your TI-
84.
 We will refer to this table when simplifying radicals
 (You can/should remove this page to make it easier for future use)

10. PRACTICE
SIMPLIFY NTH ROOTS

11. NTH ROOTS OF VARIABLES
 Remember:
 Rules of Exponents: Power of a Power Rule 𝑎 𝑚 𝑛
= 𝑎 𝑚∙𝑛
 Roots & Powers are inverse operations
 Therefore, to simplify an nth root with variables: divide the exponent by the index

12. PRACTICE
SIMPLIFY NTH ROOTS