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Mixed and Entire Radicals

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Mixed and Entire Radicals

Expressing Entire Radicals as Mixed Radicals, and vice versa

- 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)

- 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?

- 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

- √16*9 is equivalent to √16 * √9 because,
- Multiplication Property of Radicals
- n√ab = n√a * n√b, where n is a natural number, and a and b are real numbers

- 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”.

- 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

- 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

- 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

- 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

- Write the radical in simplest form, if possible.

- 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)

- 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

- 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

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