MATH 10 KHAN ACADEMY VIDEOS MATTHEW AUTH 11 Order of operations 1 The Real Numbers (11) Example 11 Worked example: Order of operations (PEMDAS) 7 2 + (7 + 3 (5 2)) 4 2 12 Rational + Irrational Example 12 Classifying numbers: rational and irrational Which of the following real numbers are irrational? 8 13 Comparing Fractions 2, π, 50, 0325, 777, 81 2 Example 13 Comparing fractions 2 (unlike denominators) compare the two fractions 21 28 and 6 14 Adding and subtracting fractions Example 14 Adding fractions with unlike denominators 10 + 1 6 2 Exponents and Radicals (12) 21 Exponents (skip everything after quiz 3) Example 21 Intro to exponents 2 3 2 3 Example 22 Simplifying square roots 5 117 Example 23 Simplifying a cube root Find 3 343 Use <, >, or = to Example 24 Exponent properties with products Use laws of exponents to rewrite (1) 6 3 6 6 (2) (3x) 3 (3) (a 3 ) 4 (4) ( 2xy 2) ( 1x 2 y ) 2 ( 3x 2 y 2) (5) 7 0 Example 25 Exponent properties with quotients Use laws of exponents to rewrite 3 (1) 4 ( 3 10 (2) (3) a 3 b 4 a 2 b ) 3 25xy 6 20y 5 x 2 1
2 MATTHEW AUTH Example 26 Powers of products and quotients (integer exponents) Use laws of exponents to rewrite (1) ( 3 8 7 3) 2 ( ) 7 2 (2) 10 4 2 22 Rational Exponents Example 27 Evaluating fractional exponents Evaluate (1) 64 1 3 (2) 64 2 3 (3) ( 8 27 ) 2 3 Example 28 Evaluating fractional exponents: negative unit-fraction Evaluate (1) 1 2 (2) ( 27) 1 3 Example 2 Evaluating fractional exponents: fractional base Evaluate (1) ( 25 (2) ( 81 256 ) 1 2 ) 1 4 Example 210 Evaluating quotient of fractional exponents Evaluate 256 7 4 Example 211 Evaluating mixed radicals and exponents Evaluate 6 1 2 ( 5 6 ) 3 Example 212 Simplify higher-index roots Simplify 6 64x 8 31 Polynomials 3 Algebraic Expressions (13) Example 31 The parts of polynomial expressions Identify the terms of 3x 2 8x+7 Example 32 Simplify polynomials Simplify 3x 2 8x + 7 + 2x 3 x 2 + 8x 3 Example 33 Adding and subtracting multiple polynomials Simplify (x 3 + 3x 6) + ( 2x 2 + x 2) (3x 4) Example 34 Multiplying monomials by polynomials Multiply 4x 2 ( 3x 2 + 25x 7 ) Example 35 Squaring binomials of the form (ax + b) 2 Simplify (7x + 10) 2 32 Factoring Example 36 More examples of factoring quatratics as (x + a)(x + b) Factor (1) x 2 + 10x + (2) x 2 + 15x + 50 (3) x 2 11x + 24 (4) x 2 + 5x 14 (5) x 2 x 56 (6) x 2 5x + 24 Example 37 Difference of squares intro Factor (1) x 2 2 4 7
MATH 10 KHAN ACADEMY VIDEOS 3 (2) y 2 25 (3) 121 b 2 Example 38 Factoring difference of squares: leading coefficient 1 Factor45x 2 125 Example 3 Perfect square factorization Factor (1) x 2 + 6x + (2) a 2 + 14a + 4 Example 310 Strategy in factoring quadratics Factor (1) 6x 2 + 3x (2) 4x 2 4x 48 (3) 3x 2 + 30x + 75 (4) 7x 2 63 (5) 2x 2 + 7x + 3 4 Rational Expressions (14) Example 41 Intro to rational expression simplification Simplify x2 +6x+5 x 2 x 2 Example 42 Simplifying rational expressions: common monomial factors Simplify (1) 14x2 +7x 14x (2) 17z3 +17z 2 34z 3 51z 2 Example 43 Multiplying rational expressions Simplify a2 4 a 2 1 a+1 a+2 Example 44 Dividing rational expressions Simplify 2p+6 p+5 10 4p+20 Example 45 Adding rational expressions: unlike denominators Simplify as one fraction 5x 2x 3 + 4x2 3x+1 Example 46 Subtracting rational expressions Find the difference Express the a 2 answer as a simplified rational expression and state the domain a+2 a 3 a 2 +4a+4 51 Solving Equations 5 Equations (15) Example 51 Linear equations with parenthesis Solve (x 6) = 3(4x + 6) for x Example 52 Worked example: number of solutions to equations Solve 8(3x + 10) = 28x 14 4x for x 52 Solving quadratics by taking square roots Example 53 Solving quadratics by taking square roots Solve (1) 2x 2 + 3 = 75 (2) 2(x + 4) 2 = 242 53 Solving quadratics by factoring Example 54 Solve the following (1) s 2 2s 35 = 0 (2) 6x 2 120x + 600 = 0 (3) (2x 3) 2 = 4x 6
4 MATTHEW AUTH 54 Completing the square Example 55 Solve (1) x 2 4x = 5 (2) 10x 2 30x 8 = 0 Example 56 Worked example: Rewriting expression by completing the square Rewrite x 2 + 16x + in the form (x + a) 2 + b Example 57 Worked example: Solving equations by completing the square Solve x 2 2x 8 = 0 by completing the square Example 58 Worked example: Completing the square (leading coefficient 1) Solve 4x 2 + 40x 300 = 0 by completing the square 55 Solving Rational Equations Example 5 Equations with one rational expression Solve 14x+4 3x 2 = 8 Example 510 Equation with one rational expression (advanced) Solve x 2 x2 4 x 2 = 4 Example 511 Equation with rational expressions Solve x2 10x+21 3x+12 = x 5 x 4 Example 512 Equation with rational expressions (example 2) Solve 2x+4 1 56 Solving square-root equations x 1 = 3 x+1 Example 513 Intro to square-root equations and extraneous solutions Solve x = 2x 6 Example 514 Solving square-root equations (basic) Solve 5x 2 8 = 2x for x Example 515 Solving square-root equations: one solution Solve 3 + 5x + 6 = 12 for x Example 516 Solving square-root equations: two solution Solve 6+3w = 2w + 12 for w Example 517 Solving square-root equations: no solutions Solve 3x 7+ 2x 1 = 0 for x 6 Inequalities (18) Example 61 Testing solutions to inequalities Test if x = 0, 1, 2 and 5 solve the following inequalities (1) x + 2 2x (2) 3x + 4 > 5x Example 62 One-step inequality examples Solve each of the following (1) 05x 75 (2) 75x 125 x (3) 3 > 10 x (4) 15 < 8 Example 63 Two-step inequalitites Solve 2 3 > 4y 8 1 3 for y
MATH 10 KHAN ACADEMY VIDEOS 5 Example 64 Inequalitites with variables on both sides Solve 3p 7 < p + for p Example 65 Inequalitites with variables on both sides (with parentheses) Solve 5x + 7 > 3(x + 1) for x Example 66 Multi-step inequalities Solve each of the following for x (1) 4x + 3 < 1 (2) 5x > 8x + 27 (3) 8x 5(4x + 1) 1 + 2(4x 3) 7 The Coordinate Plane; Graphs of Equations; Circles 71 Equation of Circle Example 71 Graph the circle (x + 5) 2 + (y 5) 2 = 4 Example 72 Graph the circle (x + 1) 2 + (y 1) 2 = 74 72 Expanded equation of a circle Example 73 Graph the circle x 2 + y 2 + 4x 4y 17 = 0 References E-mail address: mauth@ccnycunyedu