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7.1 Objective The student will be able to: Find square roots. Find cube roots. If x2 = y then x is a square root of y. In the expression 64 is the radical sign and 64 is the radicand. 1. Find the square root: 64 8 2. Find the square root: - 0.04 -0.2 3. Find the square root: ± 121 11, -11 4. Find the square root: 21 5. Find the square root: -5/9 441 25 81 Cube Roots The index of a cube root is always 3. The cube root of 64 is written as 3 64 . What does cube root mean? The cube root of a number is… …the value when multiplied by itself three times gives the original number. Cube Root Vocabulary radical sign index n x radicand Perfect Cubes If a number is a perfect cube, then you can find its exact cube root. A perfect cube is a number that can be written as the cube (raised to third power) of another number. What are Perfect Cubes? • 13 = 1 x 1 x 1 = 1 • 23 = 2 x 2 x 2 = 8 • 33 = 3 x 3 x 3 = 27 • 43 = 4 x 4 x 4 = 64 • 53 = 5 x 5 x 5 = 125 • and so on and on and on….. Examples: 3 64 4 because 4 4 4 4 3 64 4 4 4 4 3 64 because 4 3 64 Examples: 27 3 3 216 6 3 64 4 125 5 3 3 3 3 27 3 216 6 64 4 125 5 Examples: 8a 2a 3 3 3 27m 3m 12 64 y 4 y 15 4 3 5 3 3 3 8a 2a 3 27m 3m 64 y 4y 12 15 4 5 Simplify Cube Roots Not all numbers or expressions have an exact cube root as in the previous examples. If a number is NOT a perfect cube, then you might be able to SIMPLIFY it. To simplify a cube root ... 1 Write the radicand as a product of two factors, where one of the factors is a perfect cube. 2 Extract the cube root of the factor that is a perfect cube. 3 The factors that are not perfect cubes will remain as the radicand. Examples: perfect cube 1) 3 54 2) 3 640 3) 3 3 3 27 2 33 2 3 64 10 4 10 3 500a b 125 4 a a b b 7 5 3 3 6 3 3 125a b 4ab 5a b 4ab 6 3 3 2 2 3 2 2 Not all cube roots can be simplified! Example: 3 30 • 30 is not a perfect cube. • 30 does not have a perfect cube factor. 3 30 cannot be simplified! 7.2 Objective The student will be able to: use the Pythagorean Theorem What is a right triangle? hypotenuse leg right angle leg It is a triangle which has an angle that is 90 degrees. The two sides that make up the right angle are called legs. The side opposite the right angle is the hypotenuse. The Pythagorean Theorem In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a2 + b2 = c2. Note: The hypotenuse, c, is always the longest side. Find the length of the hypotenuse if 1. a = 12 and 2b = 16. 2 2 12 + 16 = c 144 + 256 = c2 400 = c2 Take the square root of both sides. 2 400 c 20 = c Find the length of the hypotenuse if 2. a = 5 and b = 7. 5 2 + 7 2 = c2 25 + 49 = c2 74 = c2 Take the square root of both sides. 74 c 2 8.60 = c 3. Find the length of the hypotenuse given a = 6 and b = 12 1. 2. 3. 4. 180 324 13.42 18 Find the length of the leg, to the nearest hundredth, if 4. a = 4 and c = 10. 42 + b2 = 102 16 + b2 = 100 Solve for b. 16 - 16 + b2 = 100 - 16 b2 = 84 2 b 84 b = 9.17 Find the length of the leg, to the nearest hundredth, if 5. c = 10 and b = 7. a2 + 72 = 102 a2 + 49 = 100 Solve for a. a2 = 100 - 49 a2 = 51 2 a 51 a = 7.14 6. Find the length of the missing side given a = 4 and c = 5 1. 2. 3. 4. 1 3 6.4 9 7. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle. 73 , 3, and 8 Which side is the biggest? The square root of 73 (= 8.5)! This must be the hypotenuse (c). Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b. Sides: 73 , 3, and 8 32 + 82 = ( 73 ) 2 9 + 64 = 73 73 = 73 Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle. 8. Determine whether the triangle is a right triangle given the sides 6, 9, and 45 1. Yes 2. No 3. Purple 7.3 Objectives The student will be able to: 1. simplify square roots, and 2. simplify radical expressions. What numbers are perfect squares? 1•1=1 2•2=4 3•3=9 4 • 4 = 16 5 • 5 = 25 6 • 6 = 36 49, 64, 81, 100, 121, 144, ... 1. Simplify 147 Find a perfect square that goes into 147. 147 7 3 2. Simplify 605 Find a perfect square that goes into 605. 11 5 Simplify 1. 2. 3. 4. 2 18 . 3 8 6 2 36 2 . . . 72 How do you simplify variables in the radical? x 7 Look at these examples and try to find the pattern… 1 x 2 x 3 x 4 x 5 x 6 x x x x x 2 x 2 x x 3 x What is the answer to x x 7 3 x ? 7 x As a general rule, divide the exponent by two. The remainder stays in the radical. 4. Simplify 49x 2 Find a perfect square that goes into 49. 7x 5. Simplify 8x 12 2x 2x 25 Simplify 1. 2. 3. 4. 3x6 3x18 6 9x 18 9x 9x 36 7. Simplify 6 · 10 Multiply the radicals. 60 4 15 4 15 2 15 8. Simplify 2 14 · 3 21 Multiply the coefficients and radicals. 6 294 6 49 6 6 49 67 6 6 42 6 9.Simplify 6 x 1. 2. 3. 4. 4x . 2 3 4 4 3x 2 x 48 4 48x . . . 3 8x How do you know when a radical problem is done? 1. No radicals can be simplified. Example: 8 2. There are no fractions in the radical. 1 Example: 4 3. There are no radicals in the denominator. Example: 1 5 10. Simplify. Whew! It simplified! 108 3 Divide the radicals. 108 3 36 6 Uh oh… There is a radical in the denominator! 8 2 11. Simplify 2 8 4 1 4 Whew! It simplified again! I hope they all are like this! 4 2 2 Uh oh… Another radical in the denominator! 12. Simplify 5 7 Uh oh… There is a fraction in the radical! Since the fraction doesn’t reduce, split the radical up. 5 7 5 7 How do I get rid of the radical in the denominator? 7 7 35 49 Multiply by the “fancy one” to make the denominator a perfect square! 35 7 7.4 Objective The student will be able to: simplify radical expressions involving addition and subtraction. 1. Simplify. 3 5+4 5-2 5 Just like when adding variables, you can only combine LIKE radicals. 5 5 2. Simplify. 6 7 - 3 - 2 7 + 4 3 Which are like radicals? 4 7 3 3 Simplify 5 2 6 2 4 2 1. 2. 3. 4. 5 2 6 2 4 2 15 2 3 2 7 2 . . . . 3. Find the perimeter of a rectangle whose length is 4 6 + 3 and whose width is 2 3 - 4. 4 6+ 3 2 3 - 4. 2 3 - 4. 4 6+ 3 Perimeter = Add all of the sides 8 6 6 3 8 4. Simplify. 4 27 - 2 48 + 2 20 Simplify each radical. 4 9 3 2 16 3 2 4 5 4 3 32 4 32 2 5 12 3 8 3 4 5 Combine like radicals. 4 34 5 5. Simplify 8 50 + 5 72 - 2 98 8 25 2 5 36 2 2 49 2 8 5 2 5 6 2 2 7 2 40 2 30 2 14 2 56 2 Simplify 5 3 4 2 3 3 1. 2. 3. 4. 5 6 2 8 . . . 3 4 2 3 3 2 34 2 34 2 . Simplify 3 12 4 27 1. 2. 3. 4. 7 39 48 3 48 6 18 3 . . . .