Chapter 6 Algebra II Honors Notes: 6.1 Evaluate nth Roots and Use Rational Exponents If b2 = a then b is an nth root of a. nth root of a: if n is even a < 0 no real nth roots a = 0 one real nth root a > 0 two real roots n √a = a 1/n if n is odd a < 0 one real nth root a = 0 one real nth root a > 0 one real nth root Simplify rational exponents using radicals Solve equations using nth roots. Powers and radicals are inverse operations. 6.2 Apply Properties of Rational Exponents Simplify expressions with rational exponents using properties of exponents. (See review of properties P. 420 and 5.1 P. 330) Properties of Radicals Product property of radicals - radicals can be factored and simplified or like radicals and be multiplied then simplified. Quotient property of radicals - fractions within a radical can simplified then find the root or find the root of numerator and denominator then simplify. All radicals must be rationalized - no radical in the denominator. (multiply fraction to remove radical in the denominator) Add and Subtract radicals - like radicals, same index, may be added or subtracted just as adding/subtracting variable terms. 6.4 Use Inverse Functions relation - pairing of input values with output values inverse relation - interchange the input and output values, (domain and range are also interchanged.) graph of in inverse relation is a reflection of the graph of the original relation over the line y = x. Find an inverse relation: 1. write the original relation equation 2. switch x and y 3. may solve for y to write in slope-intercept form Inverse functions - both the original relation and inverse are functions Horizontal line test on original graph will determine if the inverse is a function. Prove functions are inverses of each other: If f(g(x) = x g(f(x)) = x then they are inverses Example: Find the inverse and verify they are inverse functions: f(x) = x + 4 y=x+4 x=y+4 y=x-4 -1 f (x) = x - 4 f(x) = x + 4 f(g(x)) = x - 4 + 4 =x g(x) = x - 4 g (f(x)) = x + 4 - 4 =x yes they are inverses 6.5 Graph Square Root & Cube Root Functions Square Root Parent Function: f(x) = √x Domain: x ≥ 0, Range: y ≥ 0 Translation: f(x) = a√ x - h + k Cube Root Parent Function: g(x) = ∛x Domain and Range: all real numbers Translation: f(x) = a∛ x - h + k 6.6 Solve Radical Equations 1. Isolate the radical on one side of the equation, if necessary 2. Raise each side of the equation to the same power to eliminate the radical and obtain a linear, quadratic or other polynomial equation 3. Solve the polynomial equation 4. Check!