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Rational Exponents N.RN.2 – Rewrite expressions involving radicals and rational exponents using properties of exponents. N.RN.3 – Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. How to play… • When a new problem is shown, write down your answer on a slip of paper (write it BIG) • Do not show your answer to anyone! • When the teacher says “SHOWDOWN” slap your answer down for everyone to see. • Discuss your answers with your group, come to an agreement on the correct answer Showdown −24 (-3) 4 169 −2 8 2 2 5 ∙5 53 3 𝑥2 24 ∙ 3−2 2𝑥 2 = 21 … what is x? Rational: 𝟏 𝟑 , −𝟐, 𝟎. 𝟏, − … 𝟑 𝟕 Irrational: 𝝅 ≈ 𝟑. 𝟏𝟒𝟏𝟓 … 𝟐 ≈ 𝟏. 𝟒𝟏𝟒𝟐 … Integer: … − 𝟐, −𝟏, 𝟎, 𝟏, 𝟐 … Whole: 𝟎, 𝟏, 𝟐, 𝟑 … Natural: 𝟏, 𝟐, 𝟑 … Sets of Real Numbers Always, Sometimes, Never 1. 2. 3. 4. The sum of two rational numbers is rational The product of two rational numbers is a whole number The sum of a rational number and an irrational number is rational The product of a nonzero rational number and an irrational number is irrational Always, Sometimes, Never 1. 2. 3. 4. The sum of two rational numbers is rational (A) The product of two rational numbers is a whole number (S) The sum of a rational number and an irrational number is rational (N) The product of a nonzero rational number and an irrational number is irrational (A) Properties of Exponents Name Property Example Product of Powers 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 Quotient of Powers 𝑎𝑚 𝑚−𝑛 = 𝑎 𝑎𝑛 Power of a Product Power of a Quotient Power of a Power Negative Exponent 𝑎∙𝑏 𝑛 = 𝑎𝑛 ∙ 𝑏𝑛 𝑎 𝑏 𝑛 𝑎𝑚 𝑛 𝑎−𝑛 23 ∙ 22 = 25 23 = 2∙3 2 𝑎𝑛 = 𝑛 𝑏 2 2 3 = = 𝑎𝑚𝑛 23 = 1 = 𝑛 𝑎 2 2−3 = = The “nth” root of a Index 𝑛 𝑎 Radicand Properties of Radicals Name Radical of a Product Radical of a Quotient Radical of a Radical Property 𝑛 𝑎𝑏 = 𝑛 𝑛 𝑚 Example 𝑛 𝑎 = 𝑏 𝑎= 𝑎∙ 𝑛 𝑛 𝑛 𝑏 𝑎 3 9∙ 3= 75 3 𝑏 𝑛𝑚 3 3 𝑎 = 64 = Rational Exponents & Radicals Exponent Radical 1 𝑎𝑛 𝑚 𝑎𝑛 = 𝑛 1 𝑚 𝑎𝑛 Example 1. 3 𝑛 3 𝑎m 5 = Example 1 2 𝑎 or 9 = 𝑛 𝑎 𝑚 3 2 4 = Example 2. 5 1 3 3 = With your partner… 3 1 3 Explain why it makes sense that 𝑎 = 𝑎 and 3 𝑎2 = 𝑎 2 3 Example 3: Simplify each expression. Express solutions in radical form (where necessary). A) 3 𝑥𝑦 6 = B) 𝑥 ∙ 3 𝑥 = C) 𝑥 4 𝑥 = Example 4. In parts B & C, you started with an expression in radical form, converted to rational exponent form, and then converted back to radical form. Explain the purpose of each conversion. Example 5. A) 27𝑥 9 2 3 B) What is the simplified form of 27𝑥 9 2 3 How is it related to 27𝑥 9 ? 2 − 3 ? Example 6. 2 5 2 5 1 3 − 1 3 78 ∙78 5 78 = A) 4 ∙ 4 = B) 5 ∙ 5 C) 4 3 = Example 6. D) 2k 𝑘 E) 𝑥 F) 6 5 1 3 8 8 3 5 − 𝑥 𝑦 1 4 2𝑦 5 3 𝑦 10 = = 1 1 3 4 = Example 6. 2𝑥 2 + 8𝑥 2 = G) H) 3 54𝑥 4 𝑦 3 − 𝑥 2𝑥𝑦 = Exit Card 1) A) 49 = 3 C) 8 = 2) −49 = B) D) 3 −8 = What are rational and irrational numbers and how are radicals related to rational exponents?