R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
R1.4 Radicals and Rational Exponents
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Radicals and Properties of Radicals
Definition of nth Root of a Number:
Let a and b be real numbers and let n be a positive integer. If , then b is the nth root of a. If n = 2 , the root is a square root, and if n = 3, the root is a cube root.
Principal nth Root of a Number
Let a be a real number that has at least one real nth root. The principal nth root of a is the nth root that has the same sign as a.
means the principal nth root of a
Examples: 2
R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Properties of Radicals
Let a and b be real numbers such that the indicated roots are real numbers, and let m and n be positive integers.
Property
1.
2. 3. b
4.
5.
6. for even n
for odd n
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Simplifying Radicals
Even Roots:
Odd Roots:
Even Roots with Variables:
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Rationalizing Radicals
A simplified rational expression can not have a radical in the denominator. To clear the denominator of a radical, we "rationalize" the denominator. by multiplying both the numerator and the denominator by an "appropriate factor".
Rationalizing Single­Term Denominators: 1.
2.
Rationalizing Two Term Denominators: 1.
2. 6
R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
You try
1.
2.
3.
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Operations with Radicals
Multiplication and Division
Same index:
Different index: (Requires LCM)
Rewiting using rational exponents allows us to use the rules for exponents and eliminates the cumbersome process of rewriting the roots using the LCM as the new index.
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Operations with Radicals
Addition and Subtraction:
Same index: In mathematics, we only add (or subtract) like objects. If two radicals have different indices and or different radicands, we can add or subtract them only if they can be rewritten to have the same index and the same radicand.
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
Rational Exponents
Def. If a is a real number and n is a positive integer such that the principal nth root of a exists,
then
is defined to be .
If m is a positive integer that has no common factor with n, then . 11
R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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R1­4 Radicals and Rational Exponents.notebook
September 30, 2015
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