R.8 nth Roots and Rational Expressions

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Chapter R Section 8 nth Roots and Rational
Exponents

• In this section, we will
• Evaluate nth Roots
• Simplify Radical Expressions
• Add, Subtract, Multiply and
• Divide Radical Expressions
• Rationalize Denominators
• Simplify Expressions with Rational Exponents
• Factor Expressions with Radicals or Rational
  Exponents

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Recall from Review Section 2
If a is a non-negative real number, any number b,
such that is the square root of a and
is denoted If a is a non-negative real number,
any non-negative number b, such that
is the primary
square root of a and is denoted Examples
Evaluate the following by taking the square root.
The principal root of a positive number is
positive
principal root
Negative numbers do not have real square roots
R.8 nth Roots and Rational Expressions nth Roots
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• The principal nth root of a real number a, n gt 2
  an integer, symbolized by
• is defined as follows
• where and if n is even
• where a, b are any real number if n is odd
• Examples Simplify each expression.

index
radicand
radical
principal root
R.8 nth Roots and Rational Expressions nth Roots
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• Properties of Radicals Let and
  denote positive integers and let a and b
  represent real numbers. Assuming that all
  radicals are defined
• Simplifying Radicals A radical is in simplest
  form when
• No radicals appear in the denominator of a
  fraction
• The radicand cannot have any factors that are
  perfect roots (given the index)
• Examples Simplify each expression.

R.8 nth Roots and Rational Expressions Simplify
Radical Expressions
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Simplifying Radical Expressions Containing
Variables Examples Simplify each
expression. Assume that all variables are
positive.
When we divide the exponent by the index, the
remainder remains under the radical
R.8 nth Roots and Rational Expressions Simplify
Radical Expressions
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• Adding and Subtracting Radical Expressions
• simplify each radical expression
• combine all like-radicals
  (combine the
  coefficients and keep the common radical)
• Examples Simplify each expression. Assume that
  all variables are positive.

R.8 nth Roots and Rational Expressions Add,
Subtract, Multiply and Divide Radicals
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Multiplying and Dividing Radical Expressions
Examples Simplify each expression. Assume
that all variables are positive.
R.8 nth Roots and Rational Expressions Add,
Subtract, Multiply and Divide Radicals
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Examples Simplify each expression. Assume that
all variables are positive.
Vegas Rule
R.8 nth Roots and Rational Expressions Add,
Subtract, Multiply and Divide Radicals
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Rationalizing Denominators Recall that
simplifying a radical expression means that no
radicals appear in the denominator of a
fraction. Examples Simplify each expression.
Assume that all variables are positive.
R.8 nth Roots and Rational Expressions
Rationalize Denominators
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Rationalizing Binomial Denominators

example Examples Simplify each
expression. Assume that all variables are
positive.
The conjugate of the binomial a b is a b and
the conjugate of a b is a b.
R.8 nth Roots and Rational Expressions
Rationalize Denominators
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Evaluating Rational Exponents Examples
Simplify each expression.
R.8 nth Roots and Rational Expressions Simplify
Expressions with Rational Exponents
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Simplifying Expressions Containing Rational
Exponents Recall the following from Review
Section 2
and
new!
new!
R.8 nth Roots and Rational Expressions Simplify
Expressions with Rational Exponents
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Examples Simplify each expression. Express
your answer so that only positive exponents
occur. Assume that the variables are positive.
R.8 nth Roots and Rational Expressions Simplify
Expressions with Rational Exponents
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Factoring Expressions with Radicals and/or
Rational Exponents Recall that, when
factoring, we take out the GCF with the smallest
exponent in the terms. Examples Factor each
expression. Express your answer so that only
positive exponents occur.
R.8 nth Roots and Rational Expressions Factor
Expressions with Radicals/Rational Exponents
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Examples Factor each expression. Express your
answer so that only positive exponents occur.
R.8 nth Roots and Rational Expressions Factor
Expressions with Radicals/Rational Exponents
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Examples Factor each expression. Express your
answer so that only positive exponents occur.
R.8 nth Roots and Rational Expressions Factor
Expressions with Radicals/Rational Exponents
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Examples Factor each expression. Express your
answer so that only positive exponents occur.
R.8 nth Roots and Rational Expressions Factor
Expressions with Radicals/Rational Exponents
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Example The final velocity, v, of an object in
feet per second (ft/sec) after it slides down a
frictionless inclined plane of height h feet is
where
is the initial velocity in ft/sec of the
object. What is the final velocity, v, of an
object that slides down a frictionless inclined
plane of height 2 feet with an initial velocity
of 4 ft/sec?
R.8 nth Roots and Rational Expressions
Applications
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Independent Practice You learn math by doing
math. The best way to learn math is to practice,
practice, practice. The assigned homework
examples provide you with an opportunity to
practice. Be sure to complete every assigned
problem (or more if you need additional
practice). Check your answers to the
odd-numbered problems in the back of the text to
see whether you have correctly solved each
problem rework all problems that are
incorrect. Read pp. 72-76 Homework pp. 77-79
7-51 odds, 55-73 odds,

89-93 odds, 107
Dont leave your grades to chancedo your
homework!
R.8 nth Roots and Rational Expressions
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Review of Exam Policies and Procedures Page 7 of
the Student Guide and Syllabus
From Math for Artists These are the laws of
exponents and radicals in bright, cheerful, easy
to memorize colors.
R.8 nth Roots and Rational Expressions