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Exponent and Radical Rules (6.1, 6.2) Name: ___________________________________________________ Topic Definition/Rule Multiplication x a ⋅ x b = x a+b Power to a Power b ( x a ) = x ab Power of a Product n (ab ) = a nbn Zero x 0 =1 Exponents x a = x a−b xb Division ⎛ ⎜ ⎝ Power of a Quotient 1. Simplifying 2. 3. n x ⎞⎟ = x n y⎠ yn Day 20 Block: ______________ Example(s) Simplifying Exponents Step Method 1 Label all unlabeled exponents “1” 2 Take the reciprocal of the fraction and make the outside exponent positive. 3 Get rid of any inside parentheses. 4 Reduce any fractional coefficients. Move all negatives either up or down. 5 Make the exponents positive. 6 Combine all like bases. 7 Distribute the power to all exponents. Example Use properties of exponents to simplify the following expressions. f. g. i. k. h. (51/3 x1/4)3 = = 121/8 125/6 = j. (26 l. 46)-1/6 = RATIONAL EXPONENTS QUESTION: What is the square root of a number? Determine what number fits into the . a. 5 i 5 = 5 is the _____________ of 25 b. 10 i 10 = 10 is the _____________ of 100 c. x i x = _____ is the square root of ______ d. x3 i x3 = _____ is the square root of ______ e. x9 i x9 = _____ is the square root of ______ QUESTION: x What does the square root of x mean? ix = x1 is the square root of x. RULE: __________________________ FRACTIONAL EXPONENT RULE: For any real number a and integers n and m: _____________________________ Examples: a. 161/2 = =4 c. (-8)1/3 = b. 271/3 = d. (16)1/4 = = -2 = 3 =2 RATIONAL EXPONENTS Exponent (numerator) x1/2 Base (x) Root (denominator) Exponential Notation Radical Notation x1/2 x2/3 x3/4 RADICALS am/n Index a-m/n Radicand RULE: EXAMPLES: Evaluate each of the following without the use of a calculator! 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. RULE: EXAMPLES: Evaluate each of the following without the use of a calculator! 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. A negative exponent was slipped into that last problem! How did you deal with it? RULE: EXAMPLES: Evaluate each of the following without the use of a calculator! 1. 5. 9. 2. 6. 10. 3. 4. 7. 8. 11. 12. Mad Math Minute! Name: __________________________________ 2 ______ 6. 100 8. ⎛4 3⎞ ⎜⎝ x ⎟⎠ = ______ 9. x2= 11. x5= ______ 12. ⎛ ⎜⎝ ______ 14. 16 15. ⎛5 3⎞ ⎜⎝ x ⎟⎠ ______ 17. ⎛ 15 x ⎞ ⎝ ⎠ 18. 4 5. ______ = ______ ( ) 4 1 y3= 10. 49 13. x 12 = 16. 5 1 2 x3= = _____ 4. 1 ( ) ( ) x = 7. 3. ______ 1. 3x = ______ 2. 3 x = Date: __________ 2x 3 1 1 1 2 = ____ 3 = ______ 7 4x 1 2 3 = ______ = ______ ______ x 4 ⎞⎟⎠ = ______ 3 2 = ______ = ______ RADICALS Warm Up Simplify the following square root and cube root expressions 1. 3. 2. 4. EXAMPLE ONE Simplifying square roots with variables a) b) EXAMPLE TWO Rationalizing the denominator a) 5 3 b) 6 7 1+ 2 5 EXAMPLE THREE Multiplying Radical Expressions d) c) EXAMPLE FOUR Adding and Subtracting Radical Expressions a) b) c) PRACTICE 1. 2. 3. 4. 5. 6. x 11 Simplify using the exponent rules. (No decimals, keep as fractions) 1. 2. 5. 4. 7. 3. (a3b‐6)(a2b0) 8. 6. 9. 1. 5. 6. 9. 7. 12. 2. 3 15. 24 − 3 3 13. 10. 63 32 − 53 4 5 3+7 4. 14. 2 1− 4 3 EXPONENT PROPERTIES EXAMPLE FOUR Re-write the radical expression in exponential form. a) b) c) d) EXAMPLE FIVE Simplify the expression without using a calculator. a) b) c) d)