Operations on Radicals

1. Operations on Radicals Module 14 Topic 2

2. Table of Contents • Slides 3-5: Adding and Subtracting • Slide 5: Simplifying • Slides 7-10: Examples • Slides 11-14: Multiplying • Slides 15-16: Conjugates • Slides 17-19: Dividing • Slides 20-22: Rationalizing the Denominator • Slides 23-28: Practice Problems • Audio/Video and Interactive Sites • Slide 7: Gizmo • Slide 25: Interactive

3. Addition and Subtraction • Adding and Subtracting radicals is similar to adding and subtracting polynomials. • Just as you cannot combine 3x and 6y, since they are not like terms, you cannot combine radicals unless they are like radicals.

4. If asked to simplify the expression 2x + 3x, we recognize that they each share a common variable part x that makes them like terms, hence we add the coefficients and keep the variable part the same. • Likewise, if asked to simplify radicals. If they have “like” radicals then we add or subtract the coefficients and keep the like radical. Two radical expressions are said to be like radicals if they have the same index and the same radicand.

5. If the radicals in your problem are different, be sure to check to see if the radicals can be simplified. Often times, when the radicals are simplified, they become the same radical and can then be added or subtracted. Always simplify first, if possible.

6. Recall: Simplifying Radicals Divide the number under the radical. If all numbers are not prime, continue dividing. Find pairs, for a square root, under the radical and pull them out. Multiply the items you pulled out by anything in front of the radical sign. Multiply anything left under the radical . It is done!

7. Simplify: *These are not like terms, however the 12 can be simplified. Now you can simplify by using like terms. Gizmo: Operations with Radicals

8. Example: Example:

9. DO NOT ADD THE NUMBERS UNDER THE RADICAL! You may not always be able to simplify radicals. Since the radicals are not the same, and both are in their simplest form, there is no way to combine them. The answer is the same as the problem: Answer:

10. Example: Example: Example: *(Hint: P = 2L + 2W) *Neither radical can be simplified. The expression is already in simplest form.

11. Multiplying Radicals • To multiply radicals, consider the following: Multiplying Radical Expression

12. Multiplying Radicals You can simplify the product of two radicals by writing them as one radical then simplifying. Here is an example of how you can use this Multiplication property to simplify a radical expression. Example:

13. Example: Example: Multiply *Recall that to multiply polynomials you need to use the Distributive Property. (a + b)(c – d) =a(c – d) + b(c – d) =ac – ad + bc - bd

14. You should recognize these from Topic 1 Notes You should also remember the rules of exponents, when you multiply terms with the same base, you add exponents. When you divide, you subtract exponents. Putting the above information together, you should see that these rules can be combined to solve problems such as those below.

15. Conjugates Example: Distribute (2 ways to do so) Notice that just like a difference of two squares, the middle terms cancel out. We call these conjugate pairs. The conjugate of . They are a conjugate pair. When we multiply a conjugate pair, the radical cancels out and we obtain a rational number.

16. Simplified Radical Expressions are recognized by… • No radicands have perfect square factors other than 1. • No radicands contain fractions. • No radicals appear in the denominator of a fraction.

17. Dividing Radicals • To divide radicals, consider the following: Dividing Radical Expressions

18. Dividing Radicals You can simplify the quotient of two radicals by writing them as one radical then simplifying. Example:

19. Dividing Radicals Example: Notice that after simplifying the radical, we still have a square root in the denominator. We have to find a way to get rid of the radical. This is called rationalizing the denominator.

20. Rationalizing the Denominator. • Case I: There is ONE TERM in the denominator and it is a SQUARE ROOT. When the denominator is a monomial (one term), multiply both the numerator and the denominator by whatever makes the denominator an expression that can be simplified so that it no longer contains a radical. In this case it happens to be exactly the same as the denominator.

21. Case II: There is ONE TERM in the denominator, however, THE INDEX IS GREATER THAN TWO. Sometimes you need to multiply by whatever makes the denominator a perfect cube or any other power greater than 2 that can be simplified.

22. Case III: There are TWO TERMS in the denominator. We also use conjugate pairs to rationalize denominators. Be sure to enclose expressions with multiple terms in ( ). This will help you to remember to FOIL these expressions. Always reduce the root index (numbers outside radical) to the simplest form (lowest) for the final answer.

23. Application/Critical Thinking A. Find a radical expression for the perimeter and area of a right triangle with side lengths Perimeter … (P = a + b + c) Hint: In a right triangle, the longest side is the hypotenuse of the triangle. This is longer than 12, so 12 is a leg.

24. Application/Critical Thinking B. The areas of two circles are 15 square cm and 20 square cm. Find the exact ratio of the radius of the smaller circle to the radius of the larger circle. Find the radius of each first . . . A = π r2 Smaller Circle Larger Circle

25. Practice Problems Not real Practice Problems and Answers