Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems: College Algebra

Section 1.4a: Properties of Radicals




Objectives

o Roots and radical notation.o Simplifying radical expressions.




Roots and Radical Notation

nth Roots and Radical Notation

Case 1: n is an even natural number. If a is a non-negative real number and n is an even natural number, is the non-negative real number b with the property that That is In this case, note that

n a.nb a

.nn a b a b

and . n

nnn a a a a





nth Roots and Radical Notation (cont.)

Case 2: n is an odd natural number. If a is any real number and n is an odd natural number, is the real number b (whose sign will be the same as the sign of a) with the property that Again,

, and . n

n nnn na b a b a a a a .nb a

n a





nth Roots and Radical Notation (cont.)

The expression expresses the root of a in radical notation. The natural number n is called the index, a is the radicand and is called the radical sign. By convention, is usually written as

n a thn

2 .

n a

Radical Sign

Radicand

Index





Note:o When n is even, is defined only when a is non-

negative.o When n is odd, is defined for all real numbers a.o We prevent any ambiguity in the meaning of

when n is even and a is non-negative by defining to be the non-negative number whose nͭ ͪ power is a.

Ex:

n a

4 2, NOT 2.

n an a

n a




Example 1: Roots and Radical Notation

Simplify the following radicals.

a.

b.

c.

d.

3 27 3 33 27.

4 81 is not a real number, as no real number raised to the fourth power is -81.

0 0 Note: for any natural number n.0 0 0 0nn

1 1 Note: for any natural number n.1 1 1 1nn

because




Example 2: Roots and Radical Notation

Simplify the following radicals.

a.

b.

c. 6 65 15625 ( 5)

3125216

56

35 125

6 216

33 3 3 3 3( 1)

66 5 6 15625 5

In general, if n is an even natural number, for any real number a. Remember, though, that if n is an odd natural number.

nn a ann a a

because

because

because




Simplifying Radical Expressions

Simplified Radical FormA radical expression is in simplified form when:

( Use as reference, )1. The radicand contains no factor with an exponent greater than or equal to the index of the radical .

2. The radicand contains no fractions ( ).

3. The denominator contains no radical ( ).

4. The greatest common factor of the index and any exponent occurring in the radicand is 1. That is, the index and any exponent in the radicand have no common factor other than 1 ( GCF(any exponent in a, n)=1 ).

n a

(exponents in )a nxy xa





In the following properties, a and b may be taken to represent constant variable, or more complicated algebraic expressions. The letters n and m represent natural numbers.

Property1.

2.

3.

Examplen n nab a b 3 344 44 4 3 34 28 8 2 44 4 4 4y y y yx xx x

nn

n

a ab b

3 33

338 8 2

x x x

m n mna a 4 4 82256 256 256 2




Example 3: Simplifying Radical ExpressionsSimplify the following radical expressions:

a.

b.

10 53 54x y 33 3 3 23 3 2 x x y y

3 233 2x y x y

1012z 22 52 3 z 52 3z

Note that since the index is 3, we look for all of the perfect cubes in the radicand.

if n is even.nn a a




Example 3: Simplifying Radical Expressions (cont.)

Simplify the following radical expressions:

c.2

36

108xy

3 3 2

323

3 4 x

y

3 2

2

3 4xy

All perfect cubes have been brought out from under the radical, and the denominator has been rationalized.





Caution!

One common error is to rewrite

These two equations are not equal! To convince yourself of this, observe the following inequality:

as a b a b

3 4 25

9 16 9 1

7

6

7 55




Simplifying Radical ExpressionsRationalizing DenominatorsCase 1: Denominator is a single term containing a root.If the denominator is a single term containing a factor of we will take advantage of the fact that

and is a or |a|, depending on whether n is odd or even.

mn a

m n m m n m nn n n na a a a a nn a




Simplifying Radical ExpressionsRationalizing DenominatorsCase 1: Denominator is a single term containing a root.(cont.)Of course, we cannot multiply the denominator by a factor of without multiplying the numerator by the same factor, as this would change the expression. So we must multiply the fraction by .

n n m

n n m

a

a

n mn a





Rationalizing DenominatorsCase 2: Denominator consists of two terms, one or both of which are square roots.Let A + B represent the denominator of the fraction under consideration, where at least one of A and B is a square root term. We will take advantage of the fact that

Note that the exponents of 2 in the end result negate the square root (or roots) initially in the denominator.

A B A B 2 2 2 2A A B B A B A AB AB B A B





Rationalizing DenominatorsCase 2: Denominator consists of two terms, one or both of which are square roots.(cont.)Once again, remember that we cannot multiply the denominator by A – B unless we multiply the numerator by this same factor.Thus, multiply the fraction by

The factor A – B is called the conjugate radical expression of A + B.

–

–.

A BA B




Example 4: Simplifying Radical Expressions

Simplify the following radical expression:4

324xy

33

3

24xxy

3

23

22x xy

23

23

2

2

y

y

23

3 3 3

4

2

x xy

y

23 42x xyy

First, simplify the numerator and denominator. Since the index is three, we are looking for perfect cubes. Here, the perfect cubes are colored green.Next, determine what to multiply the denominator by in order to eliminate the radical. Since the index is three, we want 2 and y to have exponents of three. Remember to multiply the numerator by the same factor.





Simplify the following radical expression:

38 5

38 5

8 58 5

3 8 5

8 5

3 8 5

3

8 5

Find the conjugate radical of the denominator to eliminate the radicals. Then, multiply both the numerator and the denominator by it.





Simplify the following radical expression:

7

7xx

77

xx

77

xx

2

2

7 7 749

xx

x

7 7 7

49x xx

Since the index is two, we are looking for perfect squares. In this example, the perfect squares are colored green.

Again, find the conjugate radical of the denominator to eliminate the radicals. Then, multiply both the numerator and the denominator by it.

Note: the original expression is not real if x is negative. Since x must be positive, when we pull x out of we do not need to write , but simply x.

27xx





Rationalize the numerator in the following radical expression:

8 24x yx y

8 28 2x yx y

8 2

4 8 2x y

x y x y

2 4

4 8 2

x y

x y x y

2

8 2x y

Find the conjugate radical of the numerator to eliminate the radicals in the numerator. Then, multiply both the numerator and the denominator by it and simplify as usual.

Hawkes Learning Systems: College Algebra

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