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Accelerated CCGPS…Math II Name Rational Exponents Period_____Date_______________ What is 52? 25 What is 23? 8 What is 51? 5 What is x1? x Answers . What is (52)3? 15625 What is (2-3)2 ? 1/64 _What is x0? What is 4(¼)? 1 1 . . One way to solve equations is to think of the equation as a balance and we must work with both sides to “undo” what was done to the variable. For example, if 4x + 7 = 31, we must “undo” adding 7, so subtract 7 from each side, yielding 4x = 24. Then we must “undo” multiplying by 4, so we divide both sides by 4, yielding x = 6. Solve the following equations by “undoing”: 2 5 1 6x + 8 = 38 5(x – 8) = 50 x 5 x 3 6 2 5x 40 50 11 1 6x 30 x 5x 90 2 6 x 5 x 18 x 33 A little more vocabulary: Consider the expression 36 . What is the radicand? 36 . What is the index? 2 What is the value of 36 ? 6 Explain your thought process to get this value. What number, times itself, would produce 36. For the expression is the value of 3 3 125 , what is the radicand? 125 ? 5 125 What is the index? 3 What . Explain your thought process to get this value? What number, times itself, then times itself again, would produce 125. 4 3 27 = 4 81 = _ 3 = 2/3 3 9 Furthermore, some irrational numbers in radical form can be simplified. For example: Find these values: 24 4 6 4 3 54 3 27 2 3 27 81 = 9 6 2 6 32 33 2 and 5 8 5 2 8 2 10 10 10 16 4 16 . and (Note that we look for perfect squares/cubes.) x 2 25 If we know that x2 = 25, how do we “undo” squaring a number? Show the steps to solve this equation. If we know that x3 = 8, how do we “undo” cubing a number? Show the steps to solve this equation. 3 x 5 x3 3 x 2 8 Another way to think of solving these equations is to force the exponent of x to be 1. What would that entail? How would that look? 1 25 25 2 = That would mean that 52 1 2 5 and 3 1 8 83 = 23 1 3 2 So what would these values be? 1 49 2 = 7 2 1 2 1 64 3 7 Furthermore, since 2 83 4 3 = 1 3 4 1 100000 5 = 10 5 1 5 2 1 2 1 4 10 2 1 8 3 , what would 8 3 equal? 2 3 1 3 1 1 2 2 1 = 2 2 2 2 2 4 So find these values! 3 36 2 2 1 4 8 3 1 2 3= = 216 = 3 = 4/9 1/625 . 125 9 27 Be careful to consider what the negative sign means in an exponential position as opposed to what it means as a factor of a number! For Homework, rewrite these numbers in a different form. Change radicals to positive fractional exponents and vice versa. 1 2 1. 42 42 2. 3 15 15 3 3. 5 34 35 4. 5. 14 7. 24 1 2 35 1 5 4 4 3 16 2 3 2 6. 16 3 4 14 1 1 53 1 2 24 72 2. 3 4 3. 3 5 6 2 1 1 400 1 2 1 20 4 7 4 49 1 3 16 2 7. 49 2 1 15 5 16 2 8. 49 12 9. 64 3 4. 144 2 6. 4 1 5. 1000 3 1000 3 10 10 20 12 3 4 5 4 3 16 8. 4 3 Simplify each of the following expressions completely: 1. 6 256 2 10. 1000 3 100