Download Section 8

Section 8.1 Introduction to Roots and Radicals
Introductory Algebra, Miller/O’Neill/Hyde
Andrea Hendricks
I. Overview – Chapter 8
A. Chapter 8 will cover the topic of radicals, their operations, and their applications.
B. Chapter 8 will rely heavily on your knowledge of exponents and their properties.
II. Overview of Section 8.1 – You will learn how to
A. Define a Square Root
B. Find the Principal Square Root of a Nonnegative Real Number
C. Evaluate the Square Roots of Negative Numbers
D. Define an nth-Root
E. Find Roots of Variable Expressions
F. Translate Expressions Involving nth-Roots
G. Apply the Pythagorean Theorem
III. Define a Square Root
A. Review squaring a number
B. The reverse operation of squaring a number is to find its square roots.
C. Define a square root
The number b is a square root of the number a if b 2  a .
D. Notation
1) a is the positive square root of a and is also called the principal square root.
2)  a is the negative square root of a.
E. Find the Principal Square Root of a Nonnegative Number
Emphasize that the symbol a denotes only the positive square root.
F. Evaluate the Square Roots of Negative Numbers
Negative numbers do not have real-valued square roots since there are no real
numbers whose square equals a negative number. This topic will be discussed
further in Intermediate Algebra. For this course, we will say that the square root
of a negative number is not a real number. (Good )
IV. Define an nth-Root
A. The nth-root of a number is the reverse operation of raising a number to the nth
power.
1) The number b is the nth root of the number a if b n  a .
2) The nth root of a number a is denoted n a . n is called the index of the radical
and a is called the radicand.
3) If n is a positive even integer and a > 0, then n a is the principal (positive)
nth-root of a.
4) If n is a positive even integer and a < 0, then n a is not a real number since no
real numbers raised to an even power equals a negative number.
5) If n > 1 is a positive odd integer, then n a is the nth-root of a.
6) If n > 1 is any positive integer, then n 0  0 .
B. It is very helpful to know perfect powers for the purpose of simplifying radicals.
V. Find Roots of Variable Expressions
A. Illustrate several numerical examples to introduce this concept.
52 ,
(5) 2 ,
B. Definition of
1)
n
n
3
53 ,
3
( 5)3
an
an  a if n is a positive odd integer.
a n | a | if n is a positive even integer.
Note: If the value of a  0 , then the absolute value bars may be dropped.
If the problem states that the variables represent positive real numbers,
then the absolute value bars are not necessary.
C. It is very helpful to know perfect powers of variable expressions to simplify
radicals involving variables.
1) Perfect squares: x 2 , x 4 , x6 ,...
2) Perfect cubes: x3 , x6 , x9 ,...
3) Note: If the power is evenly divisible by n, then x n is a perfect power of n.
2)
n
VI. Translate Expressions Involving nth-Roots
 It is important to distinguish the difference between squaring a number and
the square root of a number when translating English phrases to mathematical
expressions. Squaring a number is represented as x 2 but the principal square
root of a number is represented as x .
VII. Pythagorean Theorem
A. Recall the Pythagorean Theorem is a 2  b 2  c 2 , where a and b are the legs of the
right triangle and c is the hypotenuse.
B. Principal square roots are used to solve for an unknown side of a right triangle if
the lengths of the other two sides are known.
C. Process to find missing side
1) Substitute known values into the Pythagorean Theorem and simplify each side.
2) Isolate the variable to one side of the equation.
3) Use the definition of the square root to solve for the value of the variable
discarding the negative root of the number since the side of a triangle must be
positive.
VIII. Summary – You should be able to
A. Find square roots of numbers.
B. Find nth-roots of numbers.
C. Find roots of variable expressions.
D. Translate expressions involving roots.
E. Use the Pythagorean Theorem to find missing values.