1 7.1 and 7.2 Roots and Radical Expressions and Multiplying and Dividing Radical Expressions

2 Roots and Radical Expressions Since 5 2 = 25, 5 is a square root of 25. Since (-5) 2 = 25, -5 is a square root of 25. Since (5) 3 = 125, 5 is a cube root of 125. Since (-5) 2 = -125, -5 is a cube root of Since (5) 4 = 625, 5 is a fourth root of 625. Since (-5) 4 = 625, -5 is a fourth root of 625. Since (5) 5 = 3,125, 5 is a fifth root of 3,125. And the pattern continues…….

3 Roots and Radical Expressions This pattern leads to the definition of the nth root. For any real numbers a and b, and any positive integer n, if a n = b, then a is an nth root of b. Since 2 4 = 16 and (-2) 4 = 16, both 2 and –2 are fourth roots of 16. Since there is no real number x such that x 4 = -16, -16 has no real fourth root. Since –5 is the only real number whose cube is –125, -5 is the only real root of –125.

4 Roots and Radical Expressions Type of NumberNumber of Real nth Roots when n is Even Number of Real nth Roots when n is Odd Positive NegativeNone1

5 Finding All Real Roots Find all real roots. The cube roots of 0.027, -125, 1/64 The fourth roots of 625, , 81/625 The fifth roots of 0, -1, 32 The square roots of , -1, and 36/121

6 Radicals A radical sign is used to indicate a root. The number under the radical sign is called the radicand. The index gives you the degree of the root. When a number has two real roots, the positive root is called the principal root and the radicand sign indicates the principal root.

7 Radicals Find each real – number root.

8 Radicals Find the value of the expression of x = 5 and x = -5 For any negative real number a, when n is even.

9 Radicals Simplify each radical expression.

10 Radicals Simplify each radical expression.

11 Radicals are the inverse of exponents Exponents:Radicals:

12 Simplify the Radicals

13 Rules for Simplifying Radicals Square roots can simplify if there are sets of two duplicate factors. Cube roots can simplify if there are sets of three duplicate factors. Fourth roots can simplify if there are sets of four duplicate factors. Fifth roots can simplify if there are sets of five duplicate factors. And so on and so forth….

14 Simplify the Radicals

15 Simplify the Radicals

16 Simplify the Radicals

17 Simplify the Radicals

18 Simplify the Radicals

19 Simplify the Radicals

20 Simplify the Radicals Most of the time, it is easier to divide first, then simplify later.

21 Simplify the Radicals

22 Rationalizing the Denominator It is considered bad form to have a radical in the denominator of an expression. It is necessary to do some algebra so that there is no longer a radical in the denominator. This should not be here.

23 Rationalize the Denominator To rationalize the denominator, you usually have to multiply by a fraction that is equal to one that also contains numbers that allow the offending radical to be removed. Multiply by:

24 Rationalize the Denominator Multiply by:

25 Rationalize the Denominator Divide first Multiply by:

26 Rationalize the Denominator Divide first Now multiply to rationalize the denominator.

27 Rationalize the Denominator Multiply to rationalize the denominator.

28 Rationalize the Denominator Multiply to rationalize the denominator.