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Algebra II 7.1 nth roots and rational exponents Goal 1 Evaluating nthe roots bn = a -> b = nth root of a √a°√a = a definition of square root ak 。ak = a substitute ak for √a a2k = a1 product of powers property 2k = 1 set exponents equal when bases are equal k= 1 2 √a = a1/2 solve for k. Real nth roots Let n be an integer greater than 1 and a real number If n is odd, then a has one real nth root: √a = a1/n n If n is even and a> 0, then a has two real nth roots: ± √𝑎 =±a1/n 𝑛 If n is even and a = 0, then a has one nth root: √0 = 01/n = 0 𝑛 If n is even and a < 0, then a has no real nth roots Example 1 Finding nth roots Find the indicated nth root(s) of a. a. n = 5, a = -32 b. n = 3, a = 64 Rational exponents Let a1/n be an nth root of a, and let m be a positive integer. am/n = (a1/n)m = ( √𝑎)m 𝑛 a-m/n = 1 𝑎𝑚/𝑛 = 1 1 𝑛 (𝑎 ) = 1 ( 𝑛√𝑎) ,a ≠ 0 Example 2 Evaluating expressions with rational exponents a. 165/2 b. 64-2/3 Example 3 Approximating a root with a calculator Use a graphing calculator to approximate √34 3 Example 4 Solving equations using nth roots Solve each equation. a. 6x4 = 3750 b. (x + 1)3 = 18 Goal 2 Using nth roots in real life Example 5 Evaluating a model with nth roots The rate r at which an initial deposit P will grow to a balance A in t years with interest compounded n times a year is given by the formula r = n Find r is P = $1000, A = $2000, t = 11 years and n =12. Example 6 Solving an equation using an nth root A basketball has a volume of about 455.6 cubic inches. The formula for the volume of a basketball is V = 4.18879r3. Find the radius of the basketball.