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MAT 120 Chapter 10 Section 10.1: Radical Expressions & Functions 2 The number c is a square root of a if c a . Principle square root is always a ___________________________. Simplify. 25 49 64 132 4 x8 x6 36 x 2 2 8 3 27 3 y3 3 x6 2 x 2 8x 16 x10 y12 z 4 3 The number c is a cube root of a if c a . The 3 0.0144 3 3 a c. 3 64 3 z9 8 3 27x15 MAT 120 Chapter 10 nth roots. n The number c is a nth root of a if c a . The 5 32 5 32 7 y7 6 x n 4 81 6 Section 10.2: Fractions as exponents. Review exponential Rules. Zero Exponent Rule: Product Rule: Quotient Rule: Power Rule: Negative Exponent Rule: 9 x 1 9 a c. root radicand 4 81 10 x 4 10 MAT 120 Chapter 10 Rational Exponent Rule: Fractional Exponents Convert the radicals to rational exponents. 3 x2 4 2xy 3 35 a 2b 4 Convert the expressions with rational exponents to radicals. 7 5x 1 6 1 3 3xy 3 4 Simplify by using all exponential properties and positive exponents. 1 5 3 3 3 5 x x 5 6 1 2 y 3 4 2 3 x y 4 5 2 3 1 2 MAT 120 Chapter 10 Simplifying the radical expressions by using rational exponents. 6 2x 3 5 4 4 3 x a b y 20 3 x Section 10.3: Multiplying Radicals *Multiplying Radical Expressions *Simplifying Radicals by Factoring. *Multiply and simplify….WRONG! The Product Rule for Radicals. n a n b n ab Multiply 3 2 3 7 5 3x 2 5 4 x x2 x2 4 11 4 y x 5 2 12 MAT 120 Chapter 10 Simplifying Radicals using the product rule and prime factorization. 675 x 2 y 5 72 3 11340a13b 20 135 3 1500 x 3 z11 4 432r 13t 11 3x 2 6 x 3 5 800 g 4 h8 6 256r 21t 24 4 24 x 3 y 2 4 10 x 5 y 3 Multiplying Radicals 15 6 23 25 33 20 MAT 120 Chapter 10 Section 10.4: Dividing Radicals *Dividing Radical Expressions *Simplifying Radicals by Factoring. *Divide and simplify….WRONG! The Quotient Rule for Radicals. n n a n a b b Divide 72 x 3 y 53 32 3 4 80 10 4 4 3 2x Rationalizing the Denominator. 7 2 20 15 18a 9b 5 3 5 16 3a 2b MAT 120 Chapter 10 Rationalizing the Denominator. 8x 3 12 xy 2 5 3y 4 24 x 3 y 2 5 7c 2 a 3b 2 Section 10.5: Several Radical Terms * Adding and Subtracting Radicals * Products or Quotients of 2+ Radicals * Rationalizing Denominators with 2 terms * Radicals with different roots(indices) Adding and Subtracting Radicals. Combine Like Terms rules! Combine Like Terms Combine Like Radicals ________________ 1. 1. 2. 2. Simplify 3x 5 y 3 2 y 12 x 7 180 2 3 3 5 48 MAT 120 Chapter 10 Simplify 3 54 3 16 2 2 5 3x 7 x 2 3x 2 3 12 x 73 2 x 4 y 8 6 x3 16 y 5 2 y 2 3 2 x 7 Products ( Distributive Property & F.O.I.L ) 3 2 3 x 5 4 3 25 2 6 3 8 10 3 10 3 MAT 120 Chapter 10 Rationalizing the Denominator 3 5 3 5 3 4 3x Multiply & Divide with different roots. 5 x3 3 x 4 a 2b3 3 ab 2 3 x4 x 4 a 5b 3 6 7 a 3b 4 Section 10.6: Solving Radical Equations. Principle of Powers If a = b, then an = bn for any exponent n. Use caution….false statements can become true statements! x 22 4 x 23 x y 5 4 x y 3 MAT 120 Chapter 10 We use this principle to remove radicals. 3 x 5 4 x 2 Steps to solve radical equations. 1. Isolate the radical to one side of the equation. ________ check for _______ roots! x 7 2 x 3 17 9 2. Remove the radical by the Principle of Powers. *Steps 1 & 2 may have to be repeated. 3. Solve for x. 4. Always check your answer. Solve for x. 3 x 7 1 x x7 5 2 x 53 6 3 1 MAT 120 Chapter 10 Solve for x. 2x 5 1 x 3 x5 2 x7 MAT 120 Chapter 10 Section 10.7: Formulas – Pythagorean Theorem, Special Right Triangles, Midpoint and Distance Formulas. Pythagorean Theorem. In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a2 + b2 = c2. Principle of Square Roots. Solve for the missing side. 10 90 Special Right Triangles. 45o – 45o – 90o Solve for x and y. y 6 x 8 y x MAT 120 Chapter 10 Special Right Triangles. 30o – 60o – 90o Solve for x and y. y x y 8 12 x Complete the table. The missing sides are opposite the angle in the column header. Leave your answer in radical form. 60o 45o o 30 30o 60 o 45o 90 o 9 cm. 45o 45 o 90 o 7 cm. 15 cm. 10 cm. 12 cm. 6 3 cm. 12 cm. 8 2 cm. MAT 120 Chapter 10 Distance Formula & Midpoint Formula Find the distance and the midpoint between (-3, 5) and (9, -11). If M(3, 7) is the midpoint of segment AB, find the location of B when A is located at (-6, 10). MAT 120 Chapter 10 Section 10.8: Complex Numbers. Define and simplify i to exponents. Simplify radicals with negatives. 3 4 5 20 Definition of Complex Numbers. 11 16 49 50 MAT 120 Chapter 10 Add and Subtract Complex Numbers 12 3i 2 7i Multiply Complex Numbers 2i5 3i 6 5i 9 8i 4 5i 3 2i 3 2i 3 2i Complex Conjugate Product Rule. Divide Complex Numbers _________________________________________ 2 3i 5 2i 7 4i 5i