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1. 1. Algebraic Expressions - Rules for Radicals Let’s review some concepts from Algebra 1. a a = b b If you have the same index, you can rewrite division of radical expressions to help simplify. This rule also applies in the opposite direction.
2. 2. Algebraic Expressions - Rules for Radicals Let’s review some concepts from Algebra 1. a a = b b EXAMPLE # 1 : If you have the same index, you can rewrite division of radical expressions to help simplify. This rule also applies in the opposite direction. 18 18 = = 9 =3 2 2
3. 3. Algebraic Expressions - Rules for Radicals Let’s review some concepts from Algebra 1. a a = b b EXAMPLE # 1 : a b = a ⋅b If you have the same index, you can rewrite division of radical expressions to help simplify. This rule also applies in the opposite direction. 18 18 = = 9 =3 2 2 If you have the same index, you can rewrite multiplication of radical expressions to help simplify. This rule also applies in the opposite direction.
4. 4. Algebraic Expressions - Rules for Radicals Let’s review some concepts from Algebra 1. a a = b b EXAMPLE # 1 : a b = a ⋅b EXAMPLE # 2 : If you have the same index, you can rewrite division of radical expressions to help simplify. This rule also applies in the opposite direction. 18 18 = = 9 =3 2 2 If you have the same index, you can rewrite multiplication of radical expressions to help simplify. This rule also applies in the opposite direction. 2 8 = 2 ⋅ 8 = 16 = 4
5. 5. Algebraic Expressions - Rules for Radicals n a =a m m n This rule is used to change a radical expression into an algebraic expression with a rational ( fraction ) exponent. You divide the exponent by the index ( your root ) Remember, if no root is shown, the index = 2
6. 6. Algebraic Expressions - Rules for Radicals n a =a m This rule is used to change a radical expression into an algebraic expression with a rational ( fraction ) exponent. m n You divide the exponent by the index ( your root ) Remember, if no root is shown, the index = 2 EXAMPLE # 3 : 3 x =x 2 2 3
7. 7. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m m n a a = b b a b = ab
8. 8. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 4 : m n a a = b b 7 a 2b 3 ⋅ 7 a 6b1 a b = ab
9. 9. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 4 : a a = b b m n 7 a 2b 3 ⋅ 7 a 6b1 = 49a 8b 4 a b = ab Apply this rule first along with the rule for multiplying variables…
10. 10. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 4 : a a = b b m n a b = ab 7 a 2b 3 ⋅ 7 a 6b1 = 49a 8b 4 = 49 ⋅ a 8 ⋅ b 4 Now lets split up each term so you can see it…it’s the rule in the opposite direction
11. 11. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 4 : a a = b b m n a b = ab 7 a 2b 3 ⋅ 7 a 6b1 = 49a 8b 4 = 49 ⋅ a 8 ⋅ b 4 8 2 4 2 = 7 a b = 7 a 4b 2 Find the square root of the integer and apply this rule to your variables…no index is shown so it equals two…
12. 12. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 4 : a a = b b m n a b = ab 7 a 2b 3 ⋅ 7 a 6b1 = 49a 8b 4 = 49 ⋅ a 8 ⋅ b 4 8 2 4 2 = 7 a b = 7 a 4b 2 The middle steps are not necessary if you want to mentally go from step one to the answer…
13. 13. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 5 : a a = b b m n 24m 6 n 9 6m 2 n 3 a b = ab
14. 14. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 5 : a a = b b m n 24m 6 n 9 6m 2 n 3 a b = ab Apply this rule first along with the rule for dividing variables and simplify under the radical… 24m 6 n 9 = = 4m 4 n 6 6m 2 n 3
15. 15. Algebraic Expressions - Rules for Radicals We are going to use these rules to simplify expressions. You might have to use all of them together to simplify one problem. n a =a m EXAMPLE # 5 : a a = b b m n a b = ab 24m 6 n 9 6m 2 n 3 24m 6 n 9 = = 4m 4 n 6 6m 2 n 3 4 2 6 2 = 2m n = 2m n 2 3 Now apply the index rule…
16. 16. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same.
17. 17. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12
18. 18. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 1 4 6 4 8 4 = 36 a b c First, rewrite using rational exponents…. 12 4
19. 19. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 1 4 6 4 8 4 12 4 1 4 3 2 = 36 a b c = 36 a b c 2 3 As you can see, the index of the first and second term are now different…this is not allowed
20. 20. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 1 4 3 2 = 36 a b c = 36 a b c ( ) = 6 2 1 4 2 3 3 2 a b 2c3 Rewrite 36 as 6 squared…when you apply the rule for an exponent inside raised to an exponent outside, the index will reduce… (6 ) 2 1 4 =6 2⋅ 1 4 =6 1 2
21. 21. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 2 3 3 2 1 2 a b c = 6 a b 2c3 2 3 Insert this into our problem… (6 ) 2 1 4 =6 2⋅ 1 4 =6 1 2
22. 22. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 3 2 2 3 a b c = 6 a b 2c3 2 3
23. 23. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 2 3 3 2 a b c = 6 a b 2c3 2 3 Writing the answer in radical form : 1. Any integer exponents appear outside the radical 2. Any proper fraction goes under the radical 3. Improper fractions are broken up into an integer and a remainder. The integer part goes outside the radical, the remainder goes inside the radical
24. 24. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 2 3 3 2 a b c = 6 a b 2c 3 = b 2c 3 2 3 Any integer exponents appear outside the radical
25. 25. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 2 3 3 2 a b c = 6 a b 2 c 3 = b 2 c 3 61 2 3 Any proper fraction goes under the radical. To do this, apply the rational exponent rule backwards. Only your exponent will appear…
26. 26. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 2 3 3 2 a b c = 6 a b 2c 3 = b 2c 3 6 2 3 Improper fractions are broken up into an integer and a remainder. The integer part goes outside the radical, the remainder goes inside the radical. How many 2’s divide 3 with out going over ?
27. 27. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 2 3 3 2 a b c = 6 a b 2c 3 = b 2c 3 6 2 3 Improper fractions are broken up into an integer and a remainder. The integer part goes outside the radical, the remainder goes inside the radical. How many 2’s divide 3 with out going over ? One with a remainder of one.
28. 28. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 4 36a 6b8c12 6 4 1 4 8 4 12 4 3 2 1 4 = 36 a b c = 36 a b c ( ) = 6 2 1 4 3 2 1 2 2 3 3 2 a b c = 6 a b 2 c 3 = b 2 c 3 a1 6 a1 2 3 Improper fractions are broken up into an integer and a remainder. The integer part goes outside the radical, the remainder goes inside the radical. How many 2’s divide 3 with out going over ? One with a remainder of one. The integer that divides it goes outside, the remainder goes inside…
29. 29. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 6 First, rewrite using rational exponents and reduce any fractional exponents…. 8 x 6 y15 z 3 1 6 6 6 15 6 3 6 1 6 5 2 =8 x y z =8 x y z 1 1 2
30. 30. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 6 8 x 6 y15 z 3 6 6 1 6 15 6 3 6 1 6 5 2 =8 x y z =8 x y z ( ) = 2 3 1 6 5 2 1 2 1 1 2 1 2 5 2 x y z =2 x y z 1 1 1 2 Now reduce the index of your integer….
31. 31. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 6 8 x 6 y15 z 3 6 6 1 6 15 6 3 6 1 6 5 2 =8 x y z =8 x y z = (2 3 ) 1 6 5 2 1 2 1 1 2 1 2 5 2 1 2 x y z = 2 x y z = x1 1 1 Rewrite your answer in radical form…. Any integer exponents appear outside the radical
32. 32. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 6 8 x 6 y15 z 3 6 6 1 6 15 6 3 6 1 6 5 2 =8 x y z =8 x y z = (2 3 ) 1 6 5 2 1 2 1 1 2 1 2 5 2 1 2 x y z = 2 x y z = x1 21 z1 1 1 Rewrite your answer in radical form…. Any proper fraction goes under the radical. To do this, apply the rational exponent rule backwards. Only your exponent will appear…
33. 33. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 6 8 x 6 y15 z 3 6 6 1 6 15 6 3 6 1 6 5 2 =8 x y z =8 x y z ( ) = 2 3 1 6 5 2 1 2 1 1 2 1 2 5 2 1 2 x y z = 2 x y z = x1 y 2 21 z1 y1 1 1 Rewrite your answer in radical form…. Improper fractions are broken up into an integer and a remainder. The integer part goes outside the radical, the remainder goes inside the radical. How many 2’s divide 5 with out going over ? 2 with a remainder of 1…
34. 34. Algebraic Expressions - Rules for Radicals Reducing the index – sometimes when applying the rule your fractional exponent will change its index due to reducing. When that happens, all indexes must be the same. EXAMPLE : 6 8 x 6 y15 z 3 6 6 1 6 15 6 3 6 1 6 5 2 =8 x y z =8 x y z = (2 3 ) 1 6 5 2 1 2 1 1 2 1 2 5 2 1 2 x y z = 2 x y z = x1 y 2 21 z1 y1 = xy 2 2 zy 1 1 The ones as exponents are not needed in your answer, I put them there so you could see where things were going…