Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Simplifying Radicals Index Radical n a Note: Radicand With square roots the index is not written Steps for Simplifying Square Roots 1. Factor the Radicand Completely or until you find a perfect root 2. Take out perfect roots (look for pairs) 3. Everything else (no pairs) stays under the radical Root Properties: [1] [2] 0 0 n If you have an even index, you cannot take roots of negative numbers. Roots will be positive. 4 non - real answer [3] If you have an odd index, you can take the roots of both positive and negative numbers. Roots may be both positive and negative 3 27 3 3 27 3 General Notes: [1] 16 4 [2] 16 4 [3] 16 4 4 is the principal root – 4 is the secondary root (opposite of the principal root) ±4 indicates both primary and secondary roots Example 1 [A] [C] 12 24 [B] 32 [D] 120 Example 2: Simplify [B] x10 y 6 z12 [A] x 4 y 8 [C] x 7 y13 z12 [D] x17 y14 z 9 Example 3: [A] 12 x 4 y10 [C] 30 x 7 y12 [B] 18 x10 y 7 [D] 49 x17 y15 Radicals CW Solutions [1] 28 x 4 y12 2 x 2 y 6 7 [3] 20 x y 2x y [5] 162 x14 y10 9 x7 y5 2 17 10 [7] 14 x 4 y12 x y 2 6 [9] 78 x17 y10 [11] 8 5 5x 14 x8 y 5 78 x 27 x12 y 5 3x y 6 2 3y [2] 10 256 x y 4 15 [4] 121x y [6] [8] 24 16 x 5 y12 11x2 y 7 y 45 x 4 y12 3 x y 2 6 56 x10 y17 2 x y 5 8 5 14 y [10] 225 x 4 y13 15 x2 y 6 y [12] 81x14 y11 9 x7 y 5 y Radicals Simplifying Cube Roots (and beyond) 1. Factor the radicand completely 2. Take out perfect roots (triples) Example 1 a] 3 54 b] 3 24 Example 2 a] 3 40 x9 y11 b] 3 250 x12 y10 Example 3 [A] [C] Finding Roots 16x 5 4 10 15 243a b 3 4 [B] ( x 5) [D] 3 64a12b 21 Example 4 [A] Applications Using Roots The time T in seconds that it takes a pendulum to make a complete swing back and forth is given by the formula below, where L is the length of the pendulum in feet and g is the acceleration due to gravity. Find T for a 1.5 foot pendulum. Round to the nearest 100th and g = 32 ft/sec2. L T 2 g 1.5 T 2 32 1.36 seconds Example 5 [B] Applications Using Roots The distance D in miles from an observer to the horizon over flat land or water can be estimated by the formula below, where h is the height in feet of observation. How far is the horizon for a person whose eyes are at 6 feet? Round to the nearest 100th. D 1.23 h D 1.23 6 3.01 miles Simplifying Radicals 1. Multiply radicand by radicand 2. If it’s not underneath the radical then do not multiply, write together (ex: 2 3 ) Example 1 [A] 7 Multiplying Radical Expressions 2 3 5 32 [B] 15 2 5 [C] 3 5 2 3 6 3 3 5 2 15 [D] 52 3 3 15 3 5 2 3 6 Example 2 Foil a] ( x 2)( x 4) b] (5 m )(7 m ) 35 5 m 7 m m 35 12 m m c] (6 x 2)(3 x 4) d] (3 x 2)(5 x 4) 15 x 12 x 10 x 8 15x 2 x 8 Example 3 Simplify Sums / Differences •Find common radicand •Combine like terms a] 2 12 3 27 3 b] 5 8 3 18 2 2 10 2 9 2 2 2 3 2 Example 4 [A] Adding / Subtracting Roots 2 2 3 2 5 5 [B] 6 2 3 2 4 3 3 2 4 3 [C] 3 8 2 18 32 [D] 2 12 27 8 4 3 3 3 2 2 7 32 2 SPECIAL FRACTION EXPONENT: 1 n 1 The exponent is most often used in the power of n monomials. Examples: Do you notice any other type of mathematical symbols that these special fraction exponents represent? x 1 3 3 5 z 2 1 2 2 3 x 2 1 4 2 6 x 3 1 6 3 2 y 9 8 1 4 4 1 2 2 1 N Special Fraction Exponents, a , are more commonly known as radicals in which the N value represents the root or index of the radical. Index Radical Symbol 1 Radicals: a a n n Radicand Note: The square root or ½ exponent is the most common radical and does not need to have the index written. 1 2 a a Steps for Simplifying Square Roots 1. Prime Factorization: Factor the Radicand Completely 2. Write the base of all perfect squares (PAIRS) outside of the radical as product 3. Everything else (SINGLES) stays under the radical as a product. Operations with Rational (Fraction) Exponents The same operations of when to multiply, add, subtract exponents apply with rational (fraction) exponents as did with integer (whole) exponents • Hint: Remember how to find common denominators and reduce. 1) 4) 2 3 x x a a 8 3 2) 3y 2y 6 7 2 3 7 5 5) 2x 2 3 6 5 7 3) 6) 3 2 8z z 5 6 5 b 1 4 5 2 8 Radicals CW Write in rational form. 1. 3 x5 2. 5 x7 3. x5 4. 7 x3 Write in radical form. 5. 1 x3 6. 2 x5 7. 5 x2 8. 3 x7 Radicals (Roots) and Rational Exponent Form Base Rational Exponents Property: POWER ROOT m n x n ROOT x m Base OR POWER ROOT OR ( Base) POWER x m n Example 1: Change Rational to Radical Form A] x 2 3 3 Example 2: A] 3 x 5 x 2 B] 3 x 4 5 5 C] 34 x 4 x 1 x 1 2 Change Radical to Rational Form 5 x3 B] 7 2x 9 1 7 2 x 9 7 C] 6x 6 x 5 5 2 Radicals Classwork # 1 – 4: Write in rational form. 1. 3 x 5 2. 5 8x 7 3. x5 y6 4. 7 4a 8b 3 #5 – 8: Write in radical form. 5. x 1 3 6. 7 x 2 5 7. x 5 2 8. x 3 7 Radicals Classwork #2 Determine if each pair are equivalent statements or not. 1. 3. 3 4 x and 6 4 5. x x and 5 3 x and 4 3 2 x x 5 3 3 2. 5 x 3 and x and 4. 6. 6 7 x x 5 3 18 6 and 14 x x 3 9 12 Simplifying Rational Exponents • Apply normal operations with exponents. • Convert to radical form. • Simplify the radical expression based on the index and radicand. 1. 5. 27 a 1 3 8 5 2. 8 6. 9b 2 3 2 4 3 3. 5 x2 7. 8x 5 5 7 4. 4 x 8. ab c 3 7 4 2 3 9 Radicals Classwork #3 Simplify the following expressions into simplest radical form 1. x x 4. 8 9 2. 5 9 3 5 4 4 a a 4 5 5. 2 3 3. 12 3 5 1 6 5 3 52 6. 1 3 x x 3 4 Change of Base (Index or Root) • Write the radicand in prime factorization form • REDUCE the fractions of Rational Exponents to rewrite radicals. 1. 6 27 3. 10 15 x 4 2. 4. 8 4 64 3. 3 3. 4 b b 6 6 Change of Base Practice Problems 1. 4. 6 6 25 16a 2. 6 8 27x 5. 10 3. 4 3 32a 5 6. 4 81a 2 10 4 2 36 x y z Radicals Radical Equation Equation with a variable under the radical sign Extraneous Solutions Extra solutions that do not satisfy equation Radical Equation Steps [1] Isolate the radical term (if two, the more complex) [2] Square, Cube, Fourth, etc. Both Sides [3] Solve and check for extraneous solutions Example 1 [A] Solving Radical Equations Algebraically x 1 2 4 [B] y 2 1 5 y2 6 y 2 36 y 38 Example 1 [C] x 1 5 15 x 1 10 x 1 100 x 101 [D] y 3 1 7 y 3 8 y 3 64 y 61 Radicals CW Solve Algebraically. 9. x 5 12 4 10. x 5 16 x 5 256 x 251 11. 2 x 2 11 17 2x 2 6 2 x 2 36 2 x 38 x 19 12. x 6 10 4 x 6 14 x 6 196 x 190 3 5x 1 11 17 3 5x 1 6 5x 1 2 5x 1 4 5x 5 x 1 Radicals CW Solve Algebraically. 13. 4 2x 1 5 9 4 2x 1 4 2x 1 1 2x 1 1 2x 0 x0 14. 7 4 x 1 2 16 7 4 x 1 14 4x 1 2 4x 1 4 4x 3 3 x 4 Radicals CW Solve Algebraically. 15. x 5 x 7 4 No Solution 16. x 21 1 x 12 x=4 Example 2 [A] Solving Graphically 3x 1 5 x 1 [B] 2x 3 3 2x x=½ Example 2 [C] Continued y 21 1 y 12 Y=4 [D] x 1 x 6 1 x=3 Example 3 [A] No Solutions x 15 3 x [B] 3x 2 x 4 x=Ø Example 4 [A] Misc. Equations 4 x x2 8 [B] 3x 7 x 3 x = -1, -2 x=3