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Mixed and Entire Radicals

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

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  1. Mixed and Entire Radicals Expressing Entire Radicals as Mixed Radicals, and vice versa

  2. 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)

  3. 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?

  4. 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

  5. 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”.

  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

  7. 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

  8. 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

  9. 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

  10. 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)

  11. 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

  12. 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

  13. Homework • Pg. 218 - 219 • (4-5)aceg, 7a, 9, 11acegi, 14,17,19, 21, 24

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