Mixed and Entire Radicals

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# Mixed and Entire Radicals - PowerPoint PPT Presentation

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

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)
• Expressing a mixed radical as an entire radical (limited to numerical radicands)
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
• 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
• 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 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
• 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
• Simplify the radical 3√144
• Simplify the radical 4√162
• = 23√18, 34√2
• 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
• 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
• Write the radical in simplest form, if possible.
• √26, 4√32
• Cannot be simplified, 24√2
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
• 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