Math 083 Bianco Warm Up!

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Math 083 Bianco Warm Up!

• List all the perfect squares you know.
• List all the perfect cubes you know

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What is the difference?
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Rational Exponents, Radicals, and Complex Numbers
Chapter 7
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7.1

• Radicals and Radical Functions

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Square Roots

• Opposite of squaring a number is taking the
  square root of a number.
• A number b is a square root of a number a if b2
  a.
• In order to find a square root of a, you need a
  that, when squared, equals a.

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Principal Square Roots

• Principal and Negative Square Roots
• If a is a nonnegative number, then
• is the principal or nonnegative
  square root of a

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Radicands

• Radical expression is an expression containing a
  radical sign.
• Radicand is the expression under a radical sign.
• Note that if the radicand of a square root is a
  negative number, the radical is NOT a real number.

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Radicands
Example
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Perfect Squares

• Square roots of perfect square radicands simplify
  to rational numbers (numbers that can be written
  as a quotient of integers).
• Square roots of numbers that are not perfect
  squares (like 7, 10, etc.) are irrational
  numbers.
• IF REQUESTED, you can find a decimal
  approximation for these irrational numbers.
• Otherwise, leave them in radical form.

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Perfect Square Roots

• Radicands might also contain variables and powers
  of variables.
• To avoid negative radicands, assume for this
  chapter that if a variable appears in the
  radicand, it represents positive numbers only.

Example
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Cube Roots

• Cube Root
• The cube root of a real number a is written as

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Cube Roots
Example
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nth Roots

• Other roots can be found, as well.
• The nth root of a is defined as

If the index, n, is even, the root is NOT a real
number when a is negative. If the index is odd,
the root will be a real number.
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nth Roots
Example

• Simplify the following.

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nth Roots
Example

• Simplify the following. Assume that all
  variables represent positive numbers.

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nth Roots
But we may not know whether the variable a is a
positive or negative value. Since the positive
square root must indeed be positive, we might
have to use absolute value signs to guarantee the
answer is positive.
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Finding nth Roots
If n is an even positive integer, then
If n is an odd positive integer, then
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Finding nth Roots
Simplify the following.
If we know for sure that the variables represent
positive numbers, we can write our result without
the absolute value sign.
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Finding nth Roots
Example
Simplify the following.
Since the index is odd, we dont have to force
the negative root to be a negative number. If a
or b is negative (and thus changes the sign of
the answer), thats okay.
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Evaluating Rational Functions

• We can also use function notation to represent
  rational functions.
• For example,
• Evaluating a rational function for a particular
  value involves replacing the value for the
  variable(s) involved.

Example
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Root Functions
Since every value of x that is substituted into
the equation
produces a unique value of y, the root relation
actually represents a function. The domain of the
root function when the index is even, is all
nonnegative numbers. The domain of the root
function when the index is odd, is the set of all
real numbers.
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Root Functions
We have previously worked with graphing basic
forms of functions so that you have some
familiarity with their general shape. You should
have a basic familiarity with root functions, as
well.
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Graphs of Root Functions
Example
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Graphs of Root Functions
Example
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7.2

• Rational Exponents

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Exponents with Rational Numbers

• So far, we have only worked with integer
  exponents.
• In this section, we extend exponents to rational
  numbers as a shorthand notation when using
  radicals.
• The same rules for working with exponents will
  still apply.

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Understanding a1/n

• Recall that a cube root is defined so that

However, if we let b a1/3, then
Since both values of b give us the same a,
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Using Radical Notation
Example

• Use radical notation to write the following.
  Simplify if possible.

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Understanding am/n

• If m and n are positive integers greater than 1
  with m/n in lowest terms, then

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Using Radical Notation
Example

• Use radical notation to write the following.
  Simplify if possible.

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Understanding a?m/n

• as long as a-m/n is a nonzero real number.

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Using Radical Notation
Example

• Use radical notation to write the following.
  Simplify if possible.

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Using Rules for Exponents
Example

• Use properties of exponents to simplify the
  following. Write results with only positive
  exponents.

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Using Rational Exponents
Example
Use rational exponents to write as a single
radical.
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7.3

• Simplifying Radical Expressions

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Product Rule for Radicals
Product Rule for Radicals
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Simplifying Radicals
Example

• Simplify the following radical expressions.

No perfect square factor, so the radical is
already simplified.
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Simplifying Radicals
Example

• Simplify the following radical expressions.

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Quotient Rule Radicals
Quotient Rule for Radicals
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Simplifying Radicals
Example

• Simplify the following radical expressions.

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The Distance Formula
Distance Formula

• The distance d between two points (x1,y1) and
  (x2,y2) is given by

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The Distance Formula
Example

• Find the distance between (?5, 8) and (?2, 2).

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The Midpoint Formula
Midpoint Formula

• The midpoint of the line segment whose endpoints
  are (x1,y1) and (x2,y2) is the point with
  coordinates

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The Midpoint Formula
Example

• Find the midpoint of the line segment that joins
  points P(?5, 8) and P(?2, 2).

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7.4

• Adding, Subtracting, and Multiplying Radical
  Expressions

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Sums and Differences

• Rules in the previous section allowed us to split
  radicals that had a radicand which was a product
  or a quotient.
• We can NOT split sums or differences.

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Like Radicals

• In previous chapters, weve discussed the concept
  of like terms.
• These are terms with the same variables raised to
  the same powers.
• They can be combined through addition and
  subtraction.
• Similarly, we can work with the concept of like
  radicals to combine radicals with the same
  radicand.

Like radicals are radicals with the same index
and the same radicand. Like radicals can also be
combined with addition or subtraction by using
the distributive property.
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Adding and Subtracting Radical Expressions
Example
Can not simplify
Can not simplify
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Adding and Subtracting Radical Expressions
Example

• Simplify the following radical expression.

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Adding and Subtracting Radical Expressions
Example

• Simplify the following radical expression.

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Adding and Subtracting Radical Expressions
Example

• Simplify the following radical expression.
  Assume that variables represent positive real
  numbers.

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Multiplying and Dividing Radical Expressions
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Multiplying and Dividing Radical Expressions
Example

• Simplify the following radical expressions.

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Homework

• 7.1 s
• 7.2 s
• 7.3 s
• 7.4 s
• Tuesday 7.6/7.7

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