1. 1 Chapter 5, Section 5 Roots of Real Numbers

2. 2 Simplify Radicals Finding the square root of a number and squaring a number are inverse operations. To find the square root of a number n, you have to find a number whose square is n. Since 7 2 = 49, 7 is a square root of 49. Also, since (-7) 2 = 49, -7 is also a square root of 49. Therefore,  For any real numbers a and b, if a 2 = b, then a is the square root of b.

3. 3 n th roots PowersFactorsRoots a 3 = 125(5)(5)(5) = 1255 is a cube root of 125. a 4 = 81(3)(3)(3)(3) = 813 is a fourth root of 81. a 5 = 32(2)(2)(2)(2)(2) = 322 is a fifth root of 32. a n = b(a)(a)(a)(a)…(a) = b (n factors of a) a is an n th root of b. For any real numbers a and b, and any positive integer n, if a n = b, then a is an n th root of b.

4. 4 Some vocabulary radical sign index radicand

5. 5 Facts: Some numbers have more than one real nth root. For example, 36 has two square roots, 6 and -6. When there is more than one real root, the positive root is called the principal root. When no index is given, as in, the radical sign indicates the principal square root. The symbol stands for the principal nth root of b.

6. 6 Examples Indicates the principal square root of 16. Indicates the opposite of the principal square root of 16. Indicates both square roots of 16. Indicates the principal cube root of -125. Indicates the opposite of the principal 4 th root of 81.

7. 7 Real n th roots of b,, or - nif b>0if b<0b=0 even one positive root, one negative root (example) no real roots (example) one real root, 0 (example) odd one positive root, no negative roots (example) no positive roots, one negative root (example)

8. 8 Properties of radicals

9. 9 Example 1 Simplify Because there is no index written, we know we are to find the square root. We should also always factor every coefficient completely. We know 16 = 2 4, so:

10. 10 Example 2 Simplify We should always factor EVERY coefficient. If we do this, we find that 243 = 3 5.

11. 11 Example 3 Simplify We will remember that when the radicand is negative and the index is even, there is no REAL root. We will have to hold off on this problem until we learn about IMAGINARY numbers in later chapters.