Section 5.4 The Irrational Numbers

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Section 5.4The Irrational Numbers

• Objectives
• Define the irrational numbers.
• Simplify square roots.
• Perform operations with square roots.
• Rationalize the denominator.

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Define the Irrational Numbers

• The set of irrational numbers is the set of
  numbers whose decimal representations are neither
  terminating nor repeating.
• For example, a well-known irrational number is p
  because there is no last digit in its decimal
  representation

p 3.1415926535897932384626433832795
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Square Roots

• The principal square root of a nonnegative number
  n, written , is the positive number that when
  multiplied by itself gives n.
• For example,
• because 6 6
  36.
• Notice that is a rational number because
  6 is a terminating decimal.
• Not all square roots are irrational.

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Square Roots

• A perfect square is a number that is the square
  of a whole number.
• For example, here are a few of perfect squares
• 0 02
• 1 12
• 4 22
• 9 32
• The square root of a perfect square is a whole
  number

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Simplifying Square RootsProduct Rule

• If a and b represent nonnegative numbers, then
• The square root of a product is the product of
  the square roots.
• Example Simplify, if possible
• v75 b. v500 c. v17

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Simplifying Square RootsProduct Rule

• Solution
• Because 17 has no perfect square factors (other
  than 1), v17 cannot be simplified.

25 is the greatest perfect square that is a
factor of 75.
Use the product rule.
Simplify, v25 5
100 is the greatest perfect square that is a
factor of 100.
Use the product rule.
Simplify, v100 10
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Multiplying Square Roots

• If a and b are nonnegative, then we can use the
  product rule
• to multiply square roots.
• Example Multiply a. v2 v5 b. v7 v7 c.
  v6 v12
• Solution a.
• b.
• c.

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Dividing Square RootsThe Quotient Rule

• If a and b represent nonnegative real numbers and
  b ? 0, then
• The quotient of two square roots is the square
  root of the quotient.
• Example Find the quotient a. b.
  

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Dividing Square RootsThe Quotient RuleExample
Continued

• Solution

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Adding and Subtracting Square Roots

• The number that multiplies a square root is
  called the square roots coefficient.
• For example, in 3v5, 3 is the coefficient of the
  square root.
• Square roots with the same radicand can be added
  or subtracted by adding or subtracting their
  coefficients

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Adding and Subtracting Square Roots

• Example Add or subtract as indicated
• a. b.
• Solution

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Rationalizing the Denominator

• We rationalize the denominator to rewrite the
  expression so that the denominator no longer
  contains any radicals.
• Example Rationalize the denominator
• a. b.
• Solution If we multiply numerator and
  denominator by v6, the denominator becomes v6
  v6 v36 6, which is what we want. So,

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Rationalizing the DenominatorExample Continued

• b. We can multiply the numerator and denominator
  by v5 to rationalize the denominator such that