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§ 9.3

§ 9.3. Operations with Radicals. Combining Radicals. Apples to apples, oranges to oranges,… you can only add “like” things…. Two or more radical expressions that have the same indices and the same radicands are called like radicals .

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§ 9.3

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  1. §9.3 Operations with Radicals

  2. Combining Radicals Apples to apples, oranges to oranges,… you can only add “like” things…. Two or more radical expressions that have the same indices and the same radicands are called like radicals. Like radicals can be combined under addition in exactly the same way that we combined like terms under addition. Examples of this process follow. 2 elephants + 3 elephants = 5 elephants but 5 tigers + 3 gorillas = ??? Blitzer, Introductory Algebra, 5e – Slide #2 Section 9.3

  3. Like Radicals Adding and Subtracting Like Radicals Two or more square roots can be combined using the distributive property provided that they have the same radicand. Such radicals are called like radicals. Blitzer, Introductory Algebra, 5e – Slide #3 Section 9.3

  4. Adding Radicals Add: EXAMPLE Apply the distributive property. Simplify. Blitzer, Introductory Algebra, 5e – Slide #4 Section 9.3

  5. Subtracting Radicals Subtract: EXAMPLE Split radicands into two factors so that one is the largest perfect square. Take the square root of 9 and 25. Multiply: 5(3) and 2(5). We have like radicals here and we can subtract them. Simplify. Blitzer, Introductory Algebra, 5e – Slide #5 Section 9.3

  6. Combining Radicals EXAMPLE Simplify (add or subtract) by combining like radical terms: SOLUTION In (a) note that these are like radicals, so we can combine them… So let’s see…9 of something less 4 of the same thing is 5 of that thing. Apply the distributive property. Simplify. Group like terms. Apply the distributive property. Simplify. Blitzer, Introductory Algebra, 5e – Slide #6 Section 9.3

  7. Combining Radicals Important to remember: Like radicals have the same indices and radicands. Like radicals can be added or subtracted using the distributive property. In some cases, you cannot see that radicals are “like” until you simplify them. When attempting to combine radicals, you should simplify the radicals first. Then you may see that you have like radicals that can be combined. Are we like? You don’t look like me. Yep. I’m 2 square roots of 3 and you are 5 square roots of 3. We have the same indices and radicands. We’re like! Let’s see…2 of them + 5 of them = 7 of them Blitzer, Introductory Algebra, 5e – Slide #7 Section 9.3

  8. Multiplying Radicals Radicals with more than one term are multiplied in the same way that polynomials with more than one term are multiplied. We will now consider multiplication with terms containing square roots. Blitzer, Introductory Algebra, 5e – Slide #8 Section 9.3

  9. Multiplying Radicals Multiply: EXAMPLE F.O.I.L. Multiply. Take the square root of 4. Multiply 5(2). Group like terms. Combine like terms. Blitzer, Introductory Algebra, 5e – Slide #9 Section 9.3

  10. Multiplying Radicals EXAMPLE SOLUTION Use FOIL. Multiply the radicals. Blitzer, Introductory Algebra, 5e – Slide #10 Section 9.3

  11. Multiplying Radicals CONTINUED Factor the third radicand using the greatest perfect square factor. Factor the third radicand into two radicals. Simplify. Note that after simplifying we see that we don’t have any pairs of like radicals. Then we can’t combine these radicals under addition or subtraction and we are done here. Blitzer, Introductory Algebra, 5e – Slide #11 Section 9.3

  12. Multiplying Radicals Multiply: EXAMPLE F.O.I.L. Multiply. Take the square root of 4. Add Note that here we got a real number for our answer. That’s not an accident. That happened because the two expressions we multiplied are special. They are conjugates of each other. We will now define “conjugates”. Blitzer, Introductory Algebra, 5e – Slide #12 Section 9.3

  13. Conjugates Radical expressions that involve the sum and difference of the same two terms are called conjugates. To multiply conjugates, use The following three pairs of expressions are conjugates: and and and It’s important to remember that whenever you multiply two conjugates, your result will be a real number. We will use this fact in the next section. Now, consider one more example in which we multiply conjugates. Blitzer, Introductory Algebra, 5e – Slide #13 Section 9.4

  14. Multiplying Radicals Multiply: EXAMPLE Note that we are multiplying conjugates here. F.O.I.L. Multiply. Take the square root of 4. Add Blitzer, Introductory Algebra, 5e – Slide #14 Section 9.3

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