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!