Warm up Simplify the following: 3 1 1 2 4 a) (c ) b) 2 10 (x ) c) y 3 Definition of the nth root of a number. Progression: If 7 is the square root of 49 then, 7^2 = 49 If 9^4 = 6561, then 9 is the fourth root of 6561. If 2 is the 6th root of 64, then _______________ If 8^3 = 512, then ___________________________ If b n x, then ____________________________ Or if b is the nth root of x, then ________________ If ( )^2 = 49, then ___________ is the square root of 49. Chart English Power Notation Radical Notation How many real nth roots does any real number have? Well, that depends on whether the number is positive or negative, and whether the root you are taking of the number is an even root or an odd root. For example how many real 4th roots does 81 have? To answer that question using the definition of nth roots, we are looking for some number that when raised to the fourth power it will equal 81. In algebra that looks like the equation: x 4 81 So we are really asking how many real number answers are there for x. Consider the graphs of the power function y = x^4 and the line y = 81 How many real 5th roots does 81 have? Again, Lets check out the graphs y = x^5 and y = 81 How many real 6th roots does -64 have How many real 9th roots does -64 have? To Sum up the question how many real roots does a real number have, you could answer in this way: If the root we are finding is odd! Every real number (whether positive, negative or zero) has exactly one (1) real odd root. Specific examples: The fifth root of 32 is 2 and that’s all :only 2 The cube root of -512 is -8 and that’s it : just -8 If the root we are finding is even! (2)real even roots Every positive number has exactly two Every negative number has no (0) real even roots because any real number that is raised to an even power will yield an answer greater than or equal to zero. Specific examples: The 4th roots of 625 are 5 and -5 -4096 has no real 6th roots because any number raised to the 6 th power cannot equal a negative answer.