Square Roots: The square root of a number is one of its two equal factors

1

6.2 Roots and Radicals

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Square Roots: The square root of a number is one of its two equal factors. (Using symbols: a is the square root of b if a2 = b.)

Example: The square root of 36 is 6 since 66 = 36. The square root of 36 is also – 6, since (– 6) (– 6) = 36.The positive square root is called the principle square root. We will mainly be concerned with the principle square root.

The number under the radical symbol is called the radicand. (49 and 81 are the radicands.)

Note: Negative real numbers do not have square roots because any nonzero real number is positive when squared. (No number multiplied by itself will give a negative real number.)The symbol is called a . It is used to designate the principle sradic quareal rsign oot.

49 7 Examples , : 81 9 49 is read as "the square root of 49".

Example 1. Simplify the following radical expressions.

a) 121 b) 0 c) 2525

d) 49

e) 144

Answers: 11 0 Not a real nu mber 57

12

a) 9 b) 0 Not a real nuc) mber5

d) 111

e) 3

Your Turn Problem #1Simplify the following radical expressions.

a) 81 b) 0 c) 36 121d)

225e) 9

2



3The of a number is one of its three equal factors. The symbol designatesthe cube root of a

cube numb

root er.

3

3Examples: 8 2 since 2 2 2 8 125 5 (since 5 5 5 125)

These examples are read: “the cube root of 8 is 2” “the cube root of –125 is –5”

In general, x is a cube root of of y if x3=y. Also note, the cube root of negative number is a negative number.

Example 2. Simplify the following radical expressions.

3a) 27 3b) 64 3 8c)

1253d) 343

Answers: a) 3 b) 4 c) 25

d) 7

Your Turn Problem #2

Simplify the following radical expressions.

3a) 125 3b) 8 3 27c)

64

Answers: a) 5 b) 2 c) 34

3



Thus far, we have covered square roots and cube roots. There are also nth roots: 4th roots, 5th roots, etc.

4 81

5 32

615,625

Examples:

th(4 root of 81)

th(5 root of 32)th(6 root of 15,625)

1213 8

(square root of 121)

(cube root of 8)

The numbers which designate the root is called the index #. The index # for the square root is a 2. However it is not usually written. 2121 121

nn b = a if and onlyDefi if ni ation: = b

Example 3. Simplify the following radical expressions.4a) 81 5b) 32

6c) 15,625

Answers: 4

3 (Since 3a)

81)

52

(Since 2b)

32)

65

(Since 5 15,c)

625)


4a) 256 4 16b)

815c) 32 6d) 64

Answers: a) 4 b) 23

Not a real nud) mberc) 2

Note: If the index # is even, there is no real number for the nth root of a negative number.

4



nnThis gives the property: b b. 33 27 33 8 225

25 25 32 8 33 27

A few more examples where the radicand has an exponent equal to the index.

4 423 3523

9 3 3125 5 416 2 n nThis gives another property: b b.

Properties of Radicals

1st, some examples where the radical expression is raised to a power equal to the index #.

Recall that negative numbers don’t have real number square roots. It is also true that negative numbers don’t have real number nth roots if n is an even number. For the following properties, we’ll assume that the radicand is positive for any even number index.

Let’s make some observations, then we can state another property.Writing Radical Expressions in Simplest Radical Form

36 6 and 9 4 9 4 3 2 6

100 10 and 25 4 25 4 5 2 10

33 3 3216 6 and 27 8 27 8 3 2 6

n n nWe can state the following property: ab a b

5



The factors of 40 are: 40 1 402 204 105 8

The largest perfect square is 4. So we will rewrite the square root using the 4 and 10.

2 10

40 4 10

Now replace the square root of 4 with 2 and we’re done.

Procedure: Writing Radical Expressions in Simplest Radical Form:Write the square root as a product of two square roots where one of the radicands is the largest perfect square that divides evenly into the original number. Then replace the square root with the whole number it is equal to. Leave as multiplication. Note: examples of perfect squares are 1, 4, 9, 16, 25, 36, 49, etc.

a) SimExample plify:4. 40

b) Simplify: 72 The factors of 72 are: 72 1 722 363 244 186 128 9

The largest perfect square is 36. So we will rewrite the square root using the 36 and 2.Now replace the square root of 36 with 6 and we’re done.

72 36 2

6 2

use ab a b ; 'a' is the largest perfect square

6



1. Write the prime factorization of 40.

2 10

b) Simplify: 72

Another Method for Writing Square Roots in Simplest Radical FormSome students have a difficult time with the previous method. This method is a little more writing but the process is more straightforward.

a) Simplify: Examp 40le 4. (again)

1. Find the prime factorization of the given radicand.2. Circle the pairs.3. For every pair, one of the circled numbers will be written in front of the radical. Whatever numbers are not circled stay under the radical. (Multiply if more than one number.)

40 2 2 2 5 2. Circle the pairs (only a pair of 2’s).

3. One 2 is written in front, the 2 and 5 remain inside.

72 2 2 2 3 3 1. Write the prime factorization of 72.2. Circle the pairs (pair of 2’s & 3’s).

3. The 2 & 3 are written in front, the 2 remains inside.2 3 2 6 2

Answers: 3 7 5 6 20 3 5 4 3 20 3


a) 63 b) 150 c) 5 48

7



Rewrite the 56 with 8 and 7 since 8 is a perfect cube. Then replace the cube root of 8 with 2. 33 356 8 7 32 7

Writing Cube Roots in Simplest Radical FormMethod 1. Write the cube root as a product of two cube roots where one of the radicands is the largest perfect cube that divides evenly into the original number. Then replace the cube root with the integer it is equal to. Leave as multiplication. Note: examples of perfect cubes are 1, 8, 27, 64, 125, etc.

3 SimplExample ify: . 5653b) Simplify: 270

Rewrite the 270 with 27 and 10 since 27 is a perfect cube. Then replace the cube root of 27 with 3.

33 3270 27 10 33 10

1. Write the prime factorization of 56.2. Circle the groups of 3 equal factors.3. One 2 is written in front, the 7 stays inside.

1. Write the prime factorization of 72.2. Circle the groups of 3 equal factors.

3. The 3 is written in front, the 2 & 5 stay inside.

33 56 2 2 2 7 32 7

Method 2. Again, many students have a difficult time with method 1. This method is a little more writing but the process is more straightforward.

1. Find the prime factorization of the given radicand.2. Circle the groups of 3 equal factors.3. For every group of 3, one of the circled numbers will be written in front of the radical. Whatever numbers are not circled stay under the radical. (Multiply if more than one number.) 3 a) Simplify: 56Example 5.(again)

3b) Simplify: 270

3 3270 2 3 3 3 5 33 10

8



Your Turn Problem #5Simplify the following radical expressions. (Write in simplest radical form.)

3a) 81 3b) 16 3c) 250

Answers:

3a) (81=3 3 27 3) 3b) (-16=-2 2 8 2) 3c) (250=15 2 25 2)

Simplifying Square Roots that Involve Fractions

25 5 5 5 25

From an earlier example: simplifies to , since .49 7 7 7 49

25 25 5 could also be written as .

49 749 We will now need the following property:

a a if b 0

b b

In general,n

nn

a ab b

Property for Simplifying Radical Expressions that Involve Quotients.

if b 0.

9



12 1249 49

2 37

Separate into the square root of the numerator divided by the square root of the denominator.Then simplify each (write both in simplest radical form).

12 a) SimplifyE :xample 6.

49

80b) Simplify:

121Separate into the square root of the numerator divided by the square root of the denominator.Then simplify each (write both in simplest radical form).

80 80121 121

4 511

Answers:

a) 3 5

4

5b)

4 31



45a)

1648

b) 225

10



In the last example, the denominators were perfect square roots. The numerator still contained a radical but not the denominator. A rational expression (a fraction) is not considered simplified if it contains a radical in the denominator. The process of “rationalizing the denominator” will take care of this.

Rationalizing the Denominator (Square Roots)

Observe the following: 2 2 4 27 7 49 712 12 144 12

If a square root is multiplied by itself, the result is the radicand (without square root).

Procedure: Rationalizing the denominator of a square root. (If the denominator contains a non-perfect square root)

2. Then simplify.

Next Slide

a b a a b1. Multiply by . If given , write as , then multiply by .

bb b b b

11



Separate into the square root of the numerator divided by the square root of the denominator. Then multiply the denominator by itself and multiply the numerator by the same number.

2 25 5

10

25

105

55

3 38 8

88

24 2 6

4

6486

This can be simplified differently. The denominator can be written in simplest radical form 1st before multiplying by itself. 3 3

8 2 2 2

2

6 62

6422 4

6

36 3 6

92 3

33

33

2 a) SimpExample 7. lify:

5

3b) Simplify:

8

6 c) Simplify:

3

Your Turn Problem #7 Simplify the following radical expressions.

2a)

35

b) 18

15c)

5

5d)

8

Answers: a)63

b) 106

c) 3 5 d)104

12



Rationalizing the Denominator (Cube Roots)

Observe the following: 3 3 3

33 32 4 8 29 3 27 3

A perfect cube has 3 equal factors.

If a cube root is multiplied by itself, the result is not a whole number.

3 3 35 5 25 5

1. Multiply the denominator by another cube root which will make it a perfect cube root (i.e. 8, 27, 125, etc). Whatever we multiply by the denominator, we need to multiply by the numerator.

Procedure: Rationalizing the denominator of a cube root. (If the denominator contains a non-perfect cube root)

2. The denominator should be a whole number. Write the numerator in simplest radical form. Reduce the fraction if possible.

Next Slide

13



3

3

31

27 38 18

33

3

2 23 3

3

3

9

9

3Multiply the denominator by 9.

This will make a perfect cube root.

3

6

4

33

3

3

6 2 63

22

28

3

3

2

2

32

a) SimplifExam y: pl 8. 3

e

3

6b) Simplify:

4

Answer:



3 6a)

25 3

5b)

73 5

c) 3

Answers:3

a) 305

35b

49)

7

3 4c

5)

3

The EndB.R.10-16-06

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Square Roots: The square root of a number is one of its two equal factors