Supplemental Worksheet Problems To Accompany: · PDF fileAnswer. 5) Simplify the following...

© 2008 Jason Gibson / MathTutorDVD.com The Algebra 2 Tutor Section 11 – Multiplying and Dividing Radical Expressions

Supplemental Worksheet Problems To Accompany:

The Algebra 2 Tutor

Section 11 – Multiplying and Dividing Radical Expressions

Please watch Section 11 of this DVD before working these problems.

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1) Simplify the following radical expression:

7 7


32 2


( )( )3 34 250



7 3a a


( )( )4 3 2 2


( )( )3 15y y



327 3y y


( )7 2 7+


( )3 4z z z−



( )( )2 2x x− +


2

3298x



3

2

27

75

y

x y


13) Simplify the following radical expression by rationalizing the denominator:

420


35 2−



105 5+


Question

Answer


7 7

Begin.

7 7⋅

When you multiply radicals, you simply multiply the inside of each radical together and keep the result under the radical.

49

Perform the multiplication under the radical.

7

In order to simplify further, we notice that 49 is simply ‘7’, because ‘7’ times ‘7’ equals

49.

Ans:7


Question

Answer


32 2

Begin.

32 2⋅


64


8

In order to simplify further, we notice that 64 is simply ‘8, because ‘8 times ‘8 equals

64

Ans:8


Question

Answer


( )( )3 34 250

Begin.

3 4 250⋅

When you multiply radicals, you simply multiply the inside of each radical together and keep the result under the radical. In this case we have cubed roots so the final radical is also a cubed root.

3 1000


Since we don’t readily know what 3 1000 is equal to, we draw our factor tree. Since this is a cubed root, we look for triplets of numbers under the radical. For each triplet, we pull one of the numbers out and multiply what we have pulled out.

2 5 10⋅ =

Perform the multiplication. Nothing else remains under the radical because all factors at the bottom of the tree are accounted for when we looked for our triplets. To check your work, note that 10 times 10 times 10 = 1000, as it should.

Ans:10

3 1000

4 250

2 2 10 25

2 5 5 5


Question

Answer


7 3a a

Begin.

7 3a a⋅


7 3a +

Perform the multiplication under the radical. Since we are multiplying variables that have the same base, we just add the exponents.

10a

Finalize the exponent multiplication.

We draw our factor tree. Since this is a square root, we look for pairs of numbers and/or variables under the radical. (Continued on next page)

10a

a a a a a a a a a a


a a a a a⋅ ⋅ ⋅ ⋅

For each pair, we pull one of the numbers or variables out and multiply what we have pulled out.

5a

Perform the multiplication. Nothing else remains under the radical because all factors at the bottom of the tree are accounted for when we looked for our pairs.

Ans: 5a


Question

Answer


( )( )4 3 2 2

Begin.

4 2 3 2⋅ ⋅

When you multiply radicals, you simply multiply the inside of each radical together and keep the result under the radical. Additionally, you simply multiply the numbers together on the outside of the radicals as well.

8 6

Perform the multiplications.

We next write a factor tree for the radical and look for pairs of numbers or variables in the bottom of the tree.

8 6

Since there are no pairs in our factor tree, we cannot simplify the radical any further, so we have already arrived at our answer:

Ans: 8 6

6

2 3


Question

Answer


( )( )3 15y y

Begin.

3 15 y y⋅ ⋅


245 y


We next write a factor tree for the radical and look for pairs of numbers or variables in the bottom of the tree.

45 y⋅

For every pair of “ y’s “we find in the factor tree, we pull out a single ‘y’ and multiply that by the ‘45’ that was already out in front.

45y

Do the final multiplication.

Ans: 45y

2y

y y


Question

Answer


327 3y y

Begin.

327 3 y y⋅ ⋅ ⋅


481y

Perform the multiplications. We simply add the exponents together because the last 2 terms both have ‘y’ as the base.

We next write a factor tree for the radical and look for pairs of numbers or variables in the bottom of the tree. We could continue breaking down this tree, writing “9” as “3 times “3” and we could also write 2y as “y” times “y”, but we can stop at this point in the tree because we have found two perfect pairs in our square root, so we circle those.

29 y⋅

For every pair of numbers and variables in the tree, we pull out a single one and multiply them. There is nothing that remains under the tree.

29y

Do the final multiplication.

Ans: 29y

481y

9 9 2y 2y


Question

Answer


( )7 2 7+

Begin.

2 7 7 7+ ⋅

We start by distributing the 7 into the parenthesis. When you multiply radicals, you simply multiply the inside of each radical together and keep the result under the radical. Additionally, you simply multiply the numbers together on the outside of the radicals as well.

2 7 49+


2 7 7+

The first term is already fully simplified because we can’t make 7 any simpler. This is because you can only write 7 as ‘1’ times ‘7’. For the second term, we note that 49 7= because ‘7’ times ‘7’ equals ’49’. This cannot be simplified any further so we leave it as-is. We cannot add the terms any further than this because there is nothing common between the terms, in much the same way that you must leave “x+y” as-is because the terms have nothing in common.

Ans: 2 7 7+


Question

Answer


( )3 4z z z−

Begin.

3 4 3z z z z⋅ − ⋅

We start by distributing the 3 z into the parenthesis. When you multiply radicals, you simply multiply the inside of each radical together and keep the result under the radical. Additionally, you simply multiply the numbers together on the outside of the radicals as well.

2 23 4 3z z−

Perform the multiplications. We simply add the exponents together in the terms with ‘z’ in them because the bases are the same (both have ‘z’ as the base). (continued on next page)


We write the factor tree of each of the radicals and circle the pairs in each tree.

24 2z z= 2z z=

We simplify each radical by pulling out each pair and multiplying what we have pulled from the radical.

3 2 3z z⋅ − ⋅

We rewrite the subtraction using the simplified radicals. Each term is multiplied by ‘3’ because that was the coefficient in front of the terms (see above).

6 3z z−

Perform the multiplication.

3z

Perform the subtraction.

Ans: 3 z

24z

2 2 z z

2z

z z


Question

Answer


( )( )2 2x x− +

Begin.

2 2 2 2x x x x⋅ + ⋅ − ⋅ − ⋅

First Outside Inside Last

We start by performing FOIL to multiply the two terms together the into the parenthesis. When you multiply radicals, you simply multiply the inside of each radical together and keep the result under the radical.

22 4 2x x x+ − −


22 24 xx x+ − −

The terms in red at left cancel each other because 2 2x x 0− =

24 x−

Perform the subtraction referenced in the above step.

2 x−

We could write factor trees to simplify the radicals, but in this case we can see by observation that 4 2= and 2x x= . This is because “2” times “2” equals “4” and because “x” times “x” equals 2x . We cannot simplify this further so this is the final answer.

Ans: 2 x−


Question

Answer


2

3298x

Begin.

2

3298x

When dividing a radical by another radical you can write a larger radical sign and put a fraction inside it. The numerator of this fraction is the contents of the original top radical and the bottom is the contents of the original bottom radical.

2

1649x

Simplify the fraction on the inside of the radical. We can divide the top and bottom by ‘2’ to simplify the numerator and denominator on the inside.

2

1649x

We can put the radicals back where they were to begin with in order to continue to simplify.

We write a factor tree for the radical in the numerator and denominator in order to simplify the top and bottom.

249x

7 7 x x

16

4 4


16 4= 249 7x x=

We look for pairs in our factor trees. For each radical, we simplify by pulling out one number or variable for each pair that we find.

4

7x

We rewrite our fraction using the newly simplified form of the radicals. We cannot simplify any further.

Ans: 47x


Question

Answer


3

2

27

75

y

x y

Begin.

3

2

2775

yx y

When dividing a radical by another radical you can write a larger radical sign and put a fraction inside it. The numerator of this fraction is the contents of the original top radical and the bottom is the contents of the original bottom radical.

2

2

925

yx

Simplify the fraction on the inside of the radical. We can divide the top and bottom by ‘3’ to simplify the numbers in the numerator and denominator. Since 3y is divided by we can simply subtract those exponents to arrive at

y

2y in the numerator.

2

2

9

25

y

x

We can put the radicals back where they were to begin with in order to continue to simplify.


We write a factor tree for the radical in the numerator and denominator in order to simplify the top and bottom.

29 3y y= 225 5x x=

We look for pairs in our factor trees. For each radical, we simplify by pulling out one number or variable for each pair that we find.

35

yx

We rewrite our fraction using the newly simplified form of the radicals. We cannot simplify any further.

Ans: 35

yx

29y

3 3y y

225x

5 5 x x


Question

Answer


420

Begin.

420

2020

⋅

Rationalizing the denominator simply means that you want to get rid of the radical in the denominator. In order to do this, you multiply the numerator and denominator in your original problem by the denominator.

4 2020 20⋅

Perform the multiplication in the numerator and in the denominator. If you multiply two radicals together, you extend the radical and multiply the contents on the inside.

4 20

400

Carry out the multiplications.

4 20

20

We notice that 400 20=


We write a radical tree for the radical in the numerator and look for pairs.

4 2 5

20⋅

We simplify the radical by pulling out one of the ‘2’s from the pair.

8 520

Do the multiplication in the numerator.

2 5

5

Simplify the fraction by dividing both top and bottom by ‘4’.

Ans: 2 55

20

5 4

2 2


Question

Answer


35 2−

Begin.

23

5 −5 2

25+

⋅+

If you have a radical in the denominator with two terms, as in this problem, you rationalize the denominator (get rid of the radical) by multiplying top and bottom by the “conjugate” of the denominator. A conjugate just means that you change the sign between the two terms in the denominator. In this case

5 2+ is the conjugate of 5 2− .

( )( )( )

3 5 2

5 2 5 2

+

− +


( )( )( )

3 5 2

5 2 5 2

+

− +

Carry out the multiplications. In the top we are going to distribute the ‘3’ in. In the bottom we need to do FOIL.

( ) ( ) ( ) ( )3 5 3 2

5 5 5 2 2 5 2 2+ ⋅

+ ⋅ − − ⋅

Write down the multiplications. (continued on next page)


( ) ( ) ( ) ( )3 5 6

25 2 5 2 5 4+

+ − −

Finalize the multiplications.

3 5 6

25 4+−

In the bottom, notice that ( ) ( )2 5 2 5 0− =

3 5 6

5 4+

−

In the bottom, note that 25 5=

3 5 6

1+

Perform the subtraction in the denominator.

3 5 6+

We cannot simplify this any further.

Ans: 3 5 6+


Question

Answer


105 5+

Begin.

5 555

−⋅−

If you have a radical in the denominator with two terms, as in this problem, you rationalize the denominator (get rid of the radical) by multiplying top and bottom by the “conjugate” of the denominator. A conjugate just means that you change the sign between the two terms in the denominator. In this case 5−

15

05+

5 is the conjugate of 5 2+ .

( )( )( )

10 5 5

5 5 5 5

−

+ −


( ) ( ) ( ) ( )10 5 10 5

5 5 5 5 5 5 5 5⋅ − ⋅

⋅ − ⋅ + ⋅ − ⋅

Carry out the multiplications. In the top we are going to distribute the ‘3’ in. In the bottom we need to do FOIL.

( ) ( ) ( ) ( )50 10 5

25 5 5 5 5 25−

− + −

Finalize the multiplications. (continued on next page)


( ) ( )50 10 525 25

−

−

In the bottom, notice that ( ) ( )5 5 5 5 0− + =

50 10 5

25 5−−

In the bottom, note that 25 5=

50 10 5

20−

Perform the subtraction in the denominator.

( )10 5 5

20

−

Factor out 10 in the numerator.

( )5 5

2

−

Divide top and bottom by ‘10’. We cannot simplify this any further.

Ans: 5 52−

Supplemental Worksheet Problems To Accompany: · PDF fileAnswer. 5) Simplify the following...

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