1. Chapter 8
Section 2

2. Multiplying, Dividing, and Simplifying Radicals
8.2
Multiplying, Dividing, and Simplifying Radicals
Multiply square root radicals.
Simplify radicals by using the product rule.
Simplify radicals by using the quotient rule.
Simplify radicals involving variables.
Simplify other roots. 2 3 4 5

3. Multiply square root radicals.
Objective 1
Multiply square root radicals.
Slide 8.2-3

4. Multiply square root radicals.
Product Rule for Radicals
For nonnegative real numbers a and b,
and
That is, the product of two square roots is the square root of the product, and the square root of a product is the product of the square roots.
It is important to note that the radicands not be negative numbers in the product rule. Also, in general,
Slide 8.2-4

5. Using the Product Rule to Multiply Radicals
EXAMPLE 1
Using the Product Rule to Multiply Radicals
Find each product. Assume that
Solution:
Slide 8.2-5

6. Simplify radicals by using the product rule.
Objective 2
Simplify radicals by using the product rule.
Slide 8.2-6

7. Simplify radicals by using the product rule.
A square root radical is simplified when no perfect square factor remains under the radical sign.
This can be accomplished by using the product rule:
Slide 8.2-7

8. Using the Product Rule to Simplify Radicals
EXAMPLE 2
Using the Product Rule to Simplify Radicals
Simplify each radical.
Solution:
Slide 8.2-8

9. Multiplying and Simplifying Radicals
EXAMPLE 3
Multiplying and Simplifying Radicals
Find each product and simplify.
Solution:
Slide 8.2-9

10. Simplify radicals by using the quotient rule.
Objective 3
Simplify radicals by using the quotient rule.
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11. Simplify radicals by using the quotient rule.
The quotient rule for radicals is similar to the product rule.
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12. Using the Quotient Rule to Simplify Radicals
EXAMPLE 4
Using the Quotient Rule to Simplify Radicals
Simplify each radical.
Solution:
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13. Using the Quotient Rule to Divide Radicals
EXAMPLE 5
Using the Quotient Rule to Divide Radicals
Simplify.
Solution:
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14. Using Both the Product and Quotient Rules
EXAMPLE 6
Using Both the Product and Quotient Rules
Simplify.
Solution:
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15. Simplify radicals involving variables.
Objective 4
Simplify radicals involving variables.
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16. Simplify radicals involving variables.
Radicals can also involve variables.
The square root of a squared number is always nonnegative. The absolute value is used to express this.
The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers
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17. Simplifying Radicals Involving Variables
EXAMPLE 7
Simplifying Radicals Involving Variables
Simplify each radical. Assume that all variables represent positive real numbers.
Solution:
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18. Objective 5
Simplify other roots.
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19. Properties of Radicals
Simplify other roots.
To simplify cube roots, look for factors that are perfect cubes. A perfect cube is a number with a rational cube root.
For example, , and because 4 is a rational number, 64 is a perfect cube.
Properties of Radicals
For all real number for which the indicated roots exist,
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20. Simplifying Other Roots
EXAMPLE 8
Simplifying Other Roots
Simplify each radical.
Solution:
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21. Simplify other roots. (cont’d)
Other roots of radicals involving variables can also be simplified. To simplify cube roots with variables, use the fact that for any real number a,
This is true whether a is positive or negative.
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22. Simplifying Cube Roots Involving Variables
EXAMPLE 9
Simplifying Cube Roots Involving Variables
Simplify each radical.
Solution:
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23. HL# 8.2
Book Beginning Algebra
Page 511
Exercises 45,47,48,59,70,72,75,80,90,92
Page 512
Exercises 97,100,110,116,120, 121, 123