7.1 – Roots and Radical Expressions

I. Roots and Radical Expressions 5 2 = 25, thus 5 is a square root of = 125, thus 5 is a cube root of = 625, thus 5 is a fourth root of = 3125, thus 5 is a fifth root of 3125 n th root – for any real numbers a and b and positive integer n, if a n = b, then a is a nth root of b.

THINK: BE CAREFUL!!!!THINK: BE CAREFUL!!!! 2 4 = 16 and so does (-2) 4 = 16, thus both 2 and -2 are fourth roots of 16 Whereas if x 4 = -16, there is no REAL fourth root that will produce a -16. Whereas if x 3 = -125, there is a REAL third root that will produce a -125, and it is -5.

Steps to finding nth roots:Steps to finding nth roots: –Step 1: rewrite the number and all variables underneath the radical sign as a power the same as the index (don’t forget rules of exponents) –Step 2: Take out anything raised to the index –Step 3: rewrite final answer

Example 1: find the real fifth roots of the following:Example 1: find the real fifth roots of the following: –A) 0 –B) -1 –C) -32 –D) 32 –E) -243

When finding roots from a radical sign, remember taking the square root of something is finding the second root:When finding roots from a radical sign, remember taking the square root of something is finding the second root: –√4 = (+/- 2) 2 –Radical Sign – used to indicate a root –Radicand – umber underneath the sign –Index – gives the degree of the root –When a root has two possibilities, the principal root is the positive value

Example 2: Find each real number root:Example 2: Find each real number root: A) 3 √-8 B) √(-100) C) 4 √ 81 D) 3 √-27

OBSERVE: When x = 5, then √x 2 = √25 = 5 = xWhen x = 5, then √x 2 = √25 = 5 = x When x = -5, then √x 2 = √25 = 5 ≠ xWhen x = -5, then √x 2 = √25 = 5 ≠ x

Example 3: Simplify the expressions:Example 3: Simplify the expressions: A) √4x 6 B) 3 √a 3 b 6 C) 4 √x 4 y 8 D) 3 √-27c 6 E) 2n √x 6n