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Rational Exponents & Radicals Mrs. Daniel- Algebra 1 Exponents Definition: Exponent • The exponent of a number says how many times to use that number in a multiplication. It is written as a small number to the right and above the base number. The Zero Exponent Rule • Any number (excluding zero) to the zero power is always equal to one. • Examples: 1000=1 1470=1 550 =1 Negative Power Rule Let’s Practice… 1. 5-2 2. 3 −1 4 3. (-3)-3 The One Exponent Rule • Any number (excluding zero) to the first power is always equal to that number. • Examples: a1 = a 71 = 7 531 = 53 The Power Rule (Powers to Powers) When an exponential expression is raised to a power, multiply the exponents. Try these… 1. (w4)2 3. (x3)4 2. (q2)8 Products to Powers (ab)n = anbn Distribute the exponent/power to all variables and/or coefficients. For example: (6y) 2 = (62)(y2)= 36y2 (7x3)2 = (7)2(x3)2 = 49x6 Let’s Practice… 1. (5x2)2 4. (6x4)2 2. (3wk3)3 5. (n5)2(4mn-2)3 3. (-2y)4 6. (5x2)2 The Quotient Rule Let’s Practice… Power of a Fraction Let’s Practice… 3. 3 −2 4 Untangling Exponential Expressions 1. Move expressions with negative outside exponents to bottom. 2. Distribute all outside exponents. 3. Add/Subtract to combine duplicate variables. 4. No negatives in final answer!! Let’s Practice #1 Let’s Practice #2 Let’s Practice #3 Simplifying Radicals Radical Vocab How to Simplify Radicals 1. Make a factor tree of the radicand. 2. Circle all final factor pairs. 3. All circled pairs move outside the radical and become single value. 4. Multiply all values outside radical. 5. Multiply all final factors that were not circled. Place product under radical sign. Let’s Practice… 1. 225 2. 300 Let’s Practice… 3. 1 49 4. 120 How to Simplify Cubed Radicals 1. Make a factor tree of the radicand. 2. Circle all final factor groups of three. 3. All circled groups of three move outside the radical and become single value. 4. Multiply all values outside radical. 5. Multiply all final factors that were not circled. Place product under radical sign. Let’s Practice… 3 1. 375 3 2. 64 Let’s Practice… 3 3. 81 3 4. 256 Simplifying Rational Exponents Review: Radical Vocab How to Simplify Radicals 1. Make a factor tree of the radicand. 2. Circle all final factor “nth groups”. 3. All circled “nth group” move outside the radical and become single value. 4. Multiply all values outside radical. 5. Multiple all final factors that were not circle. Place product under radical sign. Let’s Practice… Simplify: 5 1. 243 4 2. 256 3 3. 135 Code: Fractional Exponents Let’s Practice #1 Rewrite in exponential form. Simplify if possible. 5 1. 𝑥 2 6 2. 𝑥 3 5 3. 32𝑥 3 3 4. 27𝑑5 4 5. 256𝑎8 Let’s Practice #2 Rewrite as radical expressions, then simplify, if possible: 1. 12𝑎 2. 6𝑥 2 3 5 2 3. 64𝑎 4 5 Let’s Practice #3 Applications Applications Applications Applications Rational & Irrational Numbers Rational Numbers • Any number that can be expressed as the 𝑝 quotient or fraction of two integers. 𝑞 • YES: – Any integers – Any decimals that ends or repeats – Any fraction • NO: – Never ending decimals Irrational Numbers • Any number that can not be expressed as a fraction. • Usually a never-ending, non-repeating decimal. • Examples: 𝜋 2, 5 1.2658945625692…. Let’s Practice… Rational or Irrational. 1. 2. 2 17 1 3 3. 0 Will it be Rational or Irrational? Sums: Rational + Rational = Rational + Irrational = Irrational + Irrational = Products: Rational x Rational = Rational x Irrational = Irrational x Irrational =