10.4 Adding and Subtracting Radical Expressions

Simplify radical expressions involving addition and subtraction.

Objective 1

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Simplify radical expressions involving addition and subtraction.

Only radical expressions with the same index and the same radicand may

be combined. Expressions such as or cannot be

simplified by combining terms.

5 3 2233 3 2 3

Add or subtract to simplify each radical expression.

3 5 7 5 3 7 5 10 5

2 11 11 3 44 11 112 3 4 11

11 112 1 3 112

2 1 6 11

7 11

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CLASSROOM EXAMPLE 1

Adding and Subtracting Radicals

Solution:

5 12 6 75 , 0y y y 5 4 3 6 25 3y y

5 2 3 6 5 3y y

10 3 30 3y y

10 30 3y

40 3y

9 5 4 10 This expression can not be simplified any further.

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CLASSROOM EXAMPLE 1

Adding and Subtracting Radicals (cont’d)

Add or subtract to simplify each radical expression.

Solution:

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Simplify radical expressions involving addition and subtraction.

Do not confuse the product rule with combining like terms. The root of a sum

does not equal the sum of the roots. For example

since

9 16 9 16

9 16 25 5, but 9 16 3 4 7.

Add or subtract to simplify the radical expression. Assume that all variables represent positive real numbers.

4 42 32 7 162 4 42 16 2 7 81 2 4 42 2 2 7 3 2

4 44 2 21 2

425 2

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CLASSROOM EXAMPLE 2

Adding and Subtracting Radicals with Higher Indexes

Solution:

4 73 3 64p q pq 3 63 3 64p q pq pq

2 3 34pq pq pq

2 34pq pq

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CLASSROOM EXAMPLE 2

Adding and Subtracting Radicals with Higher Indexes (cont’d)

Add or subtract to simplify the radical expression. Assume that all variables represent positive real numbers.

Solution:

3 7 56 16 4 200z z 3 6 46 8 2 4 100 2z z z z

3 6 436 8 2 4 100 2z z z z

2 236 2 2 4 10 2z z z z

2 2312 2 40 2z z z z

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CLASSROOM EXAMPLE 2

Adding and Subtracting Radicals with Higher Indexes (cont’d)

Add or subtract to simplify the radical expression. Assume that all variables represent positive real numbers.

Solution:

Perform the indicated operations. Assume that all variables represent positive real numbers.

32 272 2

36 108

16 2 9 32 2

36 36 3

4 2 3 32 2

6 6 3

4 2 3

3 3

4 2 3

3

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CLASSROOM EXAMPLE 3

Adding and Subtracting Radicals with Fractions

Solution:

4 10

80 81

y y

4 10

16 5 81

y y

2 5

4 5 9

y y

3

5 5

4 5 9y

y y

3

5

4 5 9y

y

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CLASSROOM EXAMPLE 3

Adding and Subtracting Radicals with Fractions (cont’d)

Perform the indicated operations. Assume that all variables represent positive real numbers.

Solution: