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Hawkes Learning Systems: College Algebra Section 1.4a: Properties of Radicals

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Objectives o Roots and radical notation. o Simplifying radical expressions.

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Roots and Radical Notation

nth Roots and Radical Notation The expression n a expresses the nth 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, 2 is usually written as . Radical Sign Index

n

a Radicand

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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, n a is the non-negative real number b with the property that bn  a. n n That is a  b  a  b .In this case, note that

  n

a

n

 a and n a n  a.

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Roots and Radical Notation

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, n a is the real number b (whose sign will be the same as the sign of a) with the property that bn  a. Again, n

a bab , n

  n

a

n

 a and n a n  a.

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Roots and Radical Notation Note: o When n is even, n a is defined only when a is nonnegative. Ex. is not a real number. o When n is odd, n a is defined for all real numbers a. Ex. = -2 o We prevent any ambiguity in the meaning of n a when n is even and a is non-negative by defining n a to be the non-negative number whose nͭ ͪ power is a. Ex: 4  2, NOT  2.

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Example: Roots and Radical Notation Simplify the following radicals. a.

3

27  3

b.

4

81

because  3  27. 3

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

0 0

Note:

n

0  0  0n  0 for any natural number n.

d. 1  1

Note:

n

1  1  1n  1 for any natural number n.

c.

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Example: Roots and Radical Notation Simplify the following radicals. 5 125 a. 3   6 216

b.

6

 5 

6

3

125  5 because      216  6

 6 15625  5

because 56  15625  ( 5) 6

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

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Simplifying Radical Expressions Simplified Radical Form A radical expression is in simplified form when: ( Use as reference, n a ) 1. The radicand contains no factor with an exponent greater than or equal to the index of the radical (exponents in a  n) . x 2. The radicand contains no fractions ( ). yx 3. The denominator contains no radical ( ). a 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 ).

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Simplifying Radical Expressions In the following properties, a and b may be taken to represent constants, variables, or more complicated algebraic expressions. The letters n and m represent natural numbers. Property 1. Product Rule 2. Quotient Rule 3.

m n

a  mn a

n

ab  n a  n b

n

a  b

n

a n b

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Example: Simplify Radical Expressions Simplify. a) b) c) d)

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Simplifying Radical Expressions Caution! One common error is to rewrite a  b as a  b

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

=

= 5 , but

=3+4=7

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Simplifying Radical Expressions Rationalizing Denominators Case 1: Denominator is a single term containing a root. If the denominator is a single term containing a factor of n a m we will take advantage of the fact that n

a m  n a n m  n a m  a n m  n a n

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

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Simplifying Radical Expressions Rationalizing Denominators Case 1: Denominator is a single term containing a root. (cont.)

Of course, we cannot multiply the denominator by a factor of n a n m without multiplying the numerator by the same factor, as this would change the expression. So we must multiply the fraction by n a n m n

a

n m

.

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Example: Rationalize the Denominator Rationalize the Denominator. Presume variables are positive. a)

b)

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Simplifying Radical Expressions Rationalizing Denominators Case 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

 A  B  A  B   A  A  B   B  A  B   A2  AB  AB  B 2  A2  B 2 Note that the exponents of 2 in the end result negate the square root (or roots) initially in the denominator.

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Simplifying Radical Expressions Rationalizing Denominators Case 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. A– B . Thus, multiply the fraction by A– B The factor A – B is called the conjugate radical expression of A + B.

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Example: Rationalize the Denominator Rationalize the Denominator. Presume variables are positive. a)

b)

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Example: Rationalize the Numerator Rationalize the Numerator. Presume variables are positive.