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Mixed and Entire Radicals. Expressing Entire Radicals as Mixed Radicals, and vice versa. Today’s Objectives. Students will be able to demonstrate an understanding of irrational numbers by: Expressing a radical as a mixed radical in simplest form (limited to numerical radicands)

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mixed and entire radicals

Mixed and Entire Radicals

Expressing Entire Radicals as Mixed Radicals, and vice versa

today s objectives
Today’s Objectives
  • Students will be able to demonstrate an understanding of irrational numbers by:
    • Expressing a radical as a mixed radical in simplest form (limited to numerical radicands)
    • Expressing a mixed radical as an entire radical (limited to numerical radicands)
proportions
Proportions
  • Recall that we can name fractions in many different ways and they will be equivalent to each other, or proportional to each other
  • For example, all of the following fractions are equivalent to the fraction 3/12:

1/4 , 5/20 , 30/120 , 100/400

  • Why is ¼ the simplest form of 3/12?
equivalent expressions
Equivalent expressions
  • Just as with fractions, equivalent expressions for any number have the same value
  • Example:
    • √16*9 is equivalent to √16 * √9 because,
      • √16*9 = √144 = 12 and √16 * √9 = 4*3 = 12
    • Similarly, 3√8*27 is equivalent to 3√8 * 3√27 because,
      • 3√8*27 = 3√216 = 6 and 3√8 * 3√27 = 2*3 = 6
  • Multiplication Property of Radicals
  • n√ab = n√a * n√b, where n is a natural number, and a and b are real numbers
multiplication property
Multiplication Property
  • We can use this property to simplify square roots and cube roots that are not perfect squares or perfect cubes
  • We can find their factors that are perfect squares or perfect cubes
  • Example: the factors of 24 are: 1,2,3,4,6,8,12,24
    • We can simplify √24 because 24 has a perfect square factor of 4.
    • Rewrite 24 as the product of two factors, one being 4
    • √24 = √4*6 = √4*√6 = 2*√6 = 2√6
    • We can read 2√6 as “2 root 6”.
multiplication property1
Multiplication Property
  • Similarly, we can simplify 3√24 because 24 has a perfect cube factor of 8.
    • Rewrite 24 as the product of two factors, one being 8
    • 3√24 = 3√8*3 = 3√8 *3√3 = 23√3
    • We can read this as “2 cube root 3”.
  • However, we cannot simplify 4√24 because 24 has no factors that can be written as a 4th power
  • We can also use prime factorization to simplify a radical
example 1 simplifying radicals using prime factorization
Example 1) Simplifying Radicals Using Prime Factorization
  • Simplify the radical √80
  • Solution:
    • √80 = √8*10 = √2*2*2*5*2
    • = √(2*2)*(2*2)*5 = √4*√4*√5
    • =2*2*√5
    • 4√5
  • Your turn:
    • Simplify the radical 3√144
    • Simplify the radical 4√162
    • = 23√18, 34√2
multiple answers
Multiple Answers
  • Some numbers, such as 200, have more than one perfect square factor
  • The factors of 200 are: 1,2,4,5,8,10,20,25,40,50,100,200
  • Since 4, 25, and 100 are perfect squares, we can simplify √200 in three ways:
    • 2√50, 5√8, 10√2
    • 10√2 is in simplest form because the radical contains no perfect square factors other than 1.
  • So, to write a radical of index n in simplest form, we write the radicand as a product of 2 factors, one of which is the greatest perfect nth power
example 2 writing radicals in simplest form
Example 2) Writing Radicals in Simplest Form
  • Write the radical in simplest form, if possible.
    • 3√40
  • Solution:
    • Look for the perfect nth factors, where n is the index of the radical.
    • The factors of 40 are: 1,2,4,5,8,10,20,40
    • The greatest perfect cube is 8 = 2*2*2, so write 40 as 8*5.
    • 3√40 = 3√8*5 = 3√8*3√5 =
    • 23√5
  • Your turn:
    • Write the radical in simplest form, if possible.
      • √26, 4√32
      • Cannot be simplified, 24√2
mixed and entire radicals1
Mixed and Entire Radicals
  • Radicals of the form n√x such as √80, or 3√144 are entire radicals
  • Radicals of the form an√x such as 4√5, or 23√18 are mixed radicals
  • We already rewrote entire radicals as mixed radicals in Examples 1 and 2
  • Here is one more example going the opposite way (mixed radical  entire radical)
example 3 writing mixed radicals as entire radicals
Example 3) Writing Mixed Radicals as Entire Radicals
  • Write the mixed radical as an entire radical
    • 33√2
  • Solution:
    • Write 3 as: 3√3*3*3 = 3√27
    • 33√2 = 3√27 * 3√2 = 3√27*2 =
    • 3√54
  • Your turn:
    • Write each mixed radical as an entire radical.
    • 4√3, 25√2
    • √48, 5√64
review
Review
  • Multiplication Property of Radicals is:
    • n√ab = n√a * n√b, where n is a natural number, and a and b are real numbers
  • to write a radical of index n in simplest form, we write the radicand as a product of 2 factors, one of which is the greatest perfect nth power
  • Radicals of the form n√x such as √80, or 3√144 are entire radicals
  • Radicals of the form an√x such as 4√5, or 23√18 are mixed radicals
homework
Homework
  • Pg. 218 - 219
    • (4-5)aceg, 7a, 9, 11acegi, 14,17,19, 21, 24