MIXED AND ENTIRE RADICALSExpressing Entire Radicals as Mixed Radicals, and vice versa

TODAY’S OBJECTIVES

Students will be able to demonstrate an understanding of irrational numbers by:1. Expressing a radical as a mixed radical in simplest form

(limited to numerical radicands)2. 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

Your turn: Simplify the radical 3√144 Simplify the radical 4√162 = 23√18, 34√2

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

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 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

Your turn: Write each mixed radical as an entire radical. 4√3, 25√2

√48, 5√64

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

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