1. Chapter R Section 7: Radical Notation and Rational Exponents
In this section, we will…
Evaluate nth Roots
Simplify Radical Expressions
Add, Subtract, Multiply and
Divide Radical Expressions
Rationalize Denominators
Simplify Expressions with Rational Exponents
Factor Expressions with Radicals or Rational Exponents

2. Recall from Review Section 2…
If a is a non-negative real number, any number b, such that is the square root of a and is denoted
If a is a non-negative real number, any non-negative number b, such that
is the primary square root of a and is denoted
Examples: Evaluate the following by taking the square root.
The principal root of a positive number is positive
principal root:
Negative numbers do not have real # square roots

3. where a, b are any real number if n is odd
The principal nth root of a real number a, n > 2 an integer, symbolized by
is defined as follows:
where and if n is even
where a, b are any real number if n is odd
Examples: Simplify each expression.
index
radicand
radical
principal root:

4. Properties of Radicals: Let and denote positive integers and let a and b represent real numbers. Assuming that all radicals are defined:
Simplifying Radicals: A radical is in simplest form when:
No radicals appear in the denominator of a fraction
The radicand cannot have any factors that are perfect roots (given the index)
Examples: Simplify each expression.

5. Simplifying Radical Expressions Containing Variables:
Examples: Simplify each expression. Assume that all variables are positive.
When we divide the exponent by the index, the remainder remains under the radical

6. Adding and Subtracting Radical Expressions:
simplify each radical expression
combine all like-radicals (combine the coefficients and keep the common radical)
Examples: Simplify each expression. Assume that all variables are positive.

7. Multiplying and Dividing Radical Expressions:
Examples: Simplify each expression. Assume that all variables are positive.

8. Examples: Simplify each expression
Examples: Simplify each expression. Assume that all variables are positive.

9. Assignment
Page 52 Problems evens and #59

10. Rationalizing Denominators: Recall that simplifying a radical expression means that no radicals appear in the denominator of a fraction.
Examples: Simplify each expression. Assume that all variables are positive.

11. Rationalizing Binomial Denominators: example:
Examples: Simplify each expression. Assume that all variables are positive.
The conjugate of the binomial a + b is a – b and the conjugate of a – b is a + b.

12. Assignment
Page 53 Problems 66 – 86 evens

13. Evaluating Rational Exponents:
Examples: Simplify each expression.

14. Convert between radical and exponential notation…
Examples:
y5/6 =
Whenever you have an integer, try typing the problem into your calculator first.
163/4 =
8 
m5/3n7/3 =
= 173/5
= 5 1/2 • 5 1/3 =
5 5/6

15. Simplifying Expressions Containing Rational Exponents: Recall the following from Review Section 2:
Laws of Exponents: For any integers m, n (assuming no divisions by 0)
and
new!
new!

16. Examples: Simplify each expression
Examples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive.