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7.1 – Roots and Radical Expressions

7.1 – Roots and Radical Expressions. I. Roots and Radical Expressions. 5 2 = 25, thus 5 is a square root of 25 5 3 = 125, thus 5 is a cube root of 125 5 4 = 625, thus 5 is a fourth root of 625 5 5 = 3125, thus 5 is a fifth root of 3125

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7.1 – Roots and Radical Expressions

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  1. 7.1 – Roots and Radical Expressions

  2. I. Roots and Radical Expressions 52 = 25, thus 5 is a square root of 25 53 = 125, thus 5 is a cube root of 125 54 = 625, thus 5 is a fourth root of 625 55 = 3125, thus 5 is a fifth root of 3125 nth root – for any real numbers a and b and positive integer n, if an = b, then a is a nth root of b.

  3. THINK: BE CAREFUL!!!! 24 = 16 and so does (-2)4 = 16, thus both 2 and -2 are fourth roots of 16 Whereas if x4 = -16, there is no REAL fourth root that will produce a -16. Whereas if x3 = -125, there is a REAL third root that will produce a -125, and it is -5.

  4. 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

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

  6. 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

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

  8. OBSERVE: • When x = 5, then √x2 = √25 = 5 = x • When x = -5, then √x2 = √25 = 5 ≠ x

  9. Example 3: Simplify the expressions: A) √4x6 B) 3√a3b6 C) 4√x4y8 D) 3√-27c6 E) 2n √x6n

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