Unit 5, Radicals

Product and Quotient Properties of Radicals

For a0, b0,and a positive integer n: n a b n a n b a n d n ab n ab , ( b 0 ) n

Properties of Radicals

For any real number a, n an a if n is a positive even integer, and n an a if n is a positive odd integer

Definition of Rational Exponents

If n is a nonzero integer, then an n a . m 1m If m and n are integers, and n 0, then an an m n a n am .

When evaluating expressions

REMEMBER- Power/Root

Operations With Radical Expressions

When you add or subtract radical expressions, first rewrite the expressions in simplest radical form, then add or subtract.

Property: Combining Radical Expressions: Sums and Differences

ax nbx n (ab)x n ax nbx n (ab)x n

Quotient of Powers Property

m a -------- = n a

Power of a Quotient Property

m (a/b) =

Power of a Product Property

m (ab) =

Product of Powers Property

m n m+n a • a = a

Power of a Power Property

n / m\ mn \ a / = a

Solving addition or subtraction problems

-Basic: make sure it's same base! or else can't combine! -Adding or subtracting compound problems: 1. Foil ----> don't forget to distribute negative 2. Combine like bases

Basic Radical Simplifying

-Don't forget to write number of root -ABSOLUTE VALUE! ---> On even root, odd variable gets absolute value

Rationalizing

-If it's a root 2 radical, multiply by itself -If it's a root 3 or > radical, multiply by needed quality Ex. root 3 of 4 ------> multiply by root 3 of 16 -Compound radical: Multiply by conjugate (3+1) then use (3-1)

Solving multiplication and division problems

-Multiplication: Put everything under one root and simplify -Division: Do same thing but remember, division sign still in place, careful when simplifying -DONT FORGET TO RATIONALIZE!!! NEVER LEAVE ANSWER NOT IN SIMPLEST FORM

Radical inequalities --> Domain restriction

-Only solve for what's in the radical!

Zero Power Property

0 a = 1

Negative Power Property

1 -n ------ a = n

Solving Radical Expressions

1. Isolate the radical 2. Raise both sides of the equation to the same power. 3. Solve the resulting equation for the variable. 4. Check you answer, as there may be extraneous roots.

Radical Inequalities

1. Solve for domain restriction ---> Solve for value ONLY in the radical 2. Solve whole equation!

extraneous roots

CHECK WHEN ITS A EVEN ROOT!