2. Chapter R: Reference: Basic Algebraic Concepts
R.1 Review of Exponents and Polynomials
R.2 Review of Factoring
R.3 Review of Rational Expressions
R.4 Review of Negative and Rational Exponents
R.5 Review of Radicals

3. R.5 Review of Radicals Radical Notation for a1/n
If a is a real number, n is a positive integer, and a1/n is a real number, then

4. R.5 Review of Radicals
In the expression
is called a radical sign,
a is called the radicand,
n is called the index.

5. R.5 Evaluating Roots Example Evaluate each root. (a) (b) (c) Solution
(b) is not a real number.
(c)

6. R.5 Review of Radicals Radical Notation for am/n
If a is a real number, m is an integer, n is a positive integer, and is a real number, then

7. R.5 Converting from Rational Exponents to Radicals
Example Write in radical form and simplify.
(a) (b) (c)
Solution
(a)
(b)
(c)

8. R.5 Converting from Radicals to Rational Exponents
Example Write in exponential form.
(a) (b) (c)
Solution
(a) (b)
(c)

9. R.5 Review of Radicals Evaluating
If n is an even positive integer, then
If n is an odd positive integer, then

10. R.5 Using Absolute Value to Simplify Roots
Example Simplify each expression.
(a) (b) (c)
Solution
(a)
(b)
(c)

11. R.5 Review of Radicals Rules for Radicals
For all real numbers a and b, and positive integers m and n for which the indicated roots are real numbers,

12. R.5 Using the Rules for Radicals to Simplify Radical Expressions
Example Simplify each expression.
(a) (b) (c)
Solution
(a)
(b)
(c)

13. R.5 Simplifying Radicals
Simplified Radicals
An expression with radicals is simplified when the following conditions are satisfied.
1. The radicand has no factor raised to a power greater than or equal to the index.
2. The radicand has no fractions.
3. No denominator contains a radical.
4. Exponents in the radicand and the index of the radical have no common factor.
5. All indicated operations have been performed (if possible).

14. R.5 Simplifying Radicals
Example Simplify each radical.
(a) (b)
Solution
(a)
(b)

15. R.5 Simplifying Radicals by Writing Them with Rational Exponents
Example Simplify each radical.
(a) (b)
Solution
(a)
(b)

16. R.5 Adding and Subtracting Like Radicals
Example Add or subtract, as indicated. Assume all
variables represent positive real numbers.
(a) (b)
Solution
(a)

17. R.5 Adding and Subtracting Like Radicals
Solution (b)

18. R.5 Multiplying Radical Expressions
Example Find each product.
(a) (b)
Solution (a) Using FOIL,

19. R.5 Multiplying Radical Expressions
Solution (b)

20. R.5 Rationalizing Denominators
The process of simplifying a radical expression so that no denominator contains a radical is called rationalizing the denominator.
Rationalizing the denominator is accomplished by multiplying by a suitable form of 1.

21. R.5 Rationalizing Denominators
Example Rationalize each denominator.
(a) (b)
Solution
(a)
(b)

22. R.5 Rationalizing a Binomial Denominator
Example Rationalize the denominator of
Solution