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
Radicals Table of Contents Slides 3-13: Perfect Squares Slides 15-19: Rules Slides 20-22: Simplifying Radicals Slide 23: Product Property Slides 24-31: Examples and Practice Problems Slides 32-35: Perfect Cubes Slides 36-40: Nth Roots Slides 41-48: Examples and Practice Problems Slides 49-53: Solving Equations Audio/Video and Interactive Sites Slide 14: Gizmos Slide 19: Gizmo Slide 24: Gizmo Slide 27: Gizmo Slide 48: Interactive What 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, ... and so on…. Since 42 16 , 4 16 . The symbol, , is called a radical sign. Finding the square root of a number and squaring a number are inverse operations. To find the square root of a number n, you must find a number whose square is n. For example, 49 is 7, since 72 = 49. Likewise, (–7)2 = 49, so –7 is also a square root of 49. We would write the final answer as: 49 7 An expression written with a radical sign is called a radical expression. The expression written under the radical sign is called the radicand. NOTE: Every positive real number has two real number square roots. 169 13 The number 0 has just one square root, 0 itself. 0 0 Negative numbers do not have real number square roots. 4 No Real Roots When evaluating we choose the positive value of a called the principal root. Evaluate 169 13 Notice, since we are evaluating, we only use the positive answer. For any real numbers a and b, if a2 = b, then a is a square root of b. a 2 b then 7 2 49 then 112 121 then b a 49 7 121 11 Just like adding and subtracting are inverse operations, finding the square root of a number and squaring a number are inverse operations. Perfect Square 2x2=4 2 The square root of 4 is ... 2 2 4 2 Perfect Square 3x3=9 The square root of 9 is ... 3 3 3 9 3 Perfect Square The square root of 16 is ... 4 x 4 = 16 4 4 4 16 4 Perfect Square 5 5 x 5 = 25 5 Can you guess what the square root of 25 is? The square root of 25 is ... 5 25 5 This is great, But…. Do you really want to draw blocks for a problem like… 2 11 probably not! If you are given a problem like this: Find 2025 Are you going to have fun getting this answer by drawing 2025 blocks? Probably not!!!!!! 2025 45 It is easier to memorize the perfect squares up to a certain point. The following should be memorized. You will see them time and time again. x x2 x x2 0 0 10 100 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 20 400 8 64 25 625 9 81 50 2500 Gizmo: Ordering and Approximating Square Roots Gizmo: Ordering and Approximating Square Roots Quick Facts about Radicals a b To name the negative square root of a, we say 25 5 a b To indicate both square roots, use the plus/minus sign which indicates positive or negative. 25 5 3 4 7 x 1 x2 x 1 x3 x 1 x4 x 1 x7 n x 1 xn Simplifying Radicals a No Real Solution • Negative numbers do not have real number square roots. • No Real Solution 25 No Real Solution a =b This symbol represents the principal square root of a. The principal square root of a non-negative number is its nonnegative square root. 25 5 Gizmo: Square Roots Simplifying Radicals 8 99 x5 y 3 z 2 Divide the number under the radical. If all numbers are not prime, continue dividing. Find pairs, for a square root, under the radical and pull them out. Multiply the items you pulled out by anything in front of the radical sign. 8 9 11 x x x x x y y y z z 8 3 3 11 x x x x x y y y z z 8 3 3 11 x x x x x y y y z z 3 Multiply anything left under the radical . x x y 8 3 x x y z 11xy It is done! 24x2 yz 11xy z Evaluate the following: 81 99 9 To solve: Find all factors Pull out pairs (using one number to represent the pair. Multiply if needed) 100 0.25 1 4 0.5 0.5 0.5 1 1 1 2 2 2 2255 25 10 x6 x3 x3 x3 100 10 10 10 99 9 Find all real roots: 81 9 9 9 0.5 0.5 0.5 0.25 1 4 81 9 0.5 0.5 0.5 1 1 1 2 2 2 1 1 1 2 2 2 0.25 0.5 1 1 4 2 Not all numbers are perfect squares • To find the roots, you will need to simplify radial expressions in which the radicand is not a perfect square using the Product Property of Square Roots. ab a b THIS IS WHERE KNOWING THE PERFECT SQUARES IS VITAL x x2 x x2 0 0 10 100 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 20 400 8 64 25 625 9 81 50 2500 Gizmo: Simplifying Radicals Examples: A. Simplify 50 Steps Explanation 50 5 5 2 25 2 Prime Factorizat ion (5)(5) 25 - A Perfect Square 5 2 Simplify 25 B. Simplify 147 Steps Explanation 147 7 7 3 49 3 Prime Factorizat ion (7)(7) 49 - A Perfect Square 7 3 Simplify 49 xy x y The general rule for reducing the radicand is to remove any perfect powers. We are only considering square roots here, so what we are looking for is any factor that is a perfect square. In the following examples we will assume that x is positive. Gizmo: Simplifying Radicals Examples: A. Evaluate 16 x 16 x 16 x 4 4 x 4 x B. Evaluate x3 x x x x 3 2 2 xx x Although x 3 is not a perfect square, it has a factor of x 2 , which is the square of x. Examples: C. Evaluate x5 x5 x4 x x4 x x2 x2 x x2 x Here the perfect square factor is x 4 , which is the square of x 2 . D. Evaluate 8x5 8x5 4 2 x 4 x 4 x 4 2 x 2 x 2 2 x In this example we could take out a 4 and a factor of x 2 , leaving behind a 2 and one factor of x. Examples: E. 3 80 x 7 y 3 3 2 2 2 2 5 xxxxxxx yyy 3 (2 2 2) 2 5 ( xxx)( xxx) x ( yyy) *All the sets of “3” have been grouped. They are cubes! 2x2 y 2x2 y 25 x 3 3 10 x Unless otherwise stated, when simplifying expressions using variables, we must use absolute value signs. n an a when n is even. NOTE: No absolute value signs are needed when finding cube roots, because a real number has just one cube root. The cube root of a positive number is positive. The cube root of a negative number is negative. Evaluate the following: 16 No real roots 49x 4 7 7 x2 x2 7 x2 1 3 2 x y 8 9 255m 1 1 1 x x x y y xy x 4 4 4 9 9 5 51 m m m m m 2 2 2 9m 2m2 255m 2 9m 4 255m What are 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….. Cubes 5 7 6 8 1 3 2 4 2x2x2=8 2 2 2 8 3 2 3 x 3 x 3 = 27 3 3 3 27 3 3 th N Roots When there is no index number, n, it is understood to be a 2 or square root. For example: x = principal square root of x. Not every radical is a square root. If there is an index number n other than the number 2, then you have a root other than a square root. th N Roots • Since 32 = 9. we call 3 the square root of 9. 9 3 • Since 33 =27 we call 3 the cube root of 27. 3 27 3 • Since 34 = 81, we call 3 the fourth root of 81. 4 81 3 • This leads us to the definition of the nth root of a number. If an = b then a is the nth root b notated as, a n b . More Explanation of Roots th N Roots • Since (-)(-) = + and (+)(+) = + , then all positive real numbers have two square roots. • Remember in our Real Number System the b is not defined. • However we can find the cube root of negative numbers since (-)(-)(-) = a negative and (+)(+)(+) = a positive. • Therefore, cube roots only have one root. Nth Roots Type of Number + Number of Real nth Roots when n is even 2 Number of Real nth Roots when n is odd. 1 0 1 1 - None 1 Nth Roots of Variables • Lets use a table to see the pattern when simplifying nth roots of variables. x x x2 3 3 n x 4 x 3 x 6 x m x x x x 2 3 3 2 x x2 x x x x x x x x x2 3 3 x m n *Note: In the first row above, the absolute value of x yields the principal root in the event that x is negative. Examples: A. Find all real cube roots of -125, 64, 0 and 9. Solutions : - 5, 4, 0 and 3 9 B. Find all real fourth roots of 16, 625, -1 and 0. Solutions : 2, 5, Undefined and 0 As previously stated when a number has two real roots, the positive root is called the principal root and the Radical indicates the principal root. Therefore when asked to find the nth root of a number we always choose the principal root. F. Simplfy 3 1000 x 3 y 9 3 1000 x y 10 x ( y ) 3 9 3 3 3 3 3 3 (10 xy3 ) 3 10xy3 Write each factor as a cube. Write as the cube of a product. Simplify. Absolute Value signs are NOT needed here because the index, n, is odd. Application/Critical Thinking 4 A. The formula for the volume of a sphere is V 3 . r Find the radius, to the nearest hundredth, of a sphere with a volume of . in 3 15 B. A student visiting the Sears Tower Skydeck is 1353 feet above the ground. Find the distance the student can see to the horizon. Use the formula d 1.5h to the approximate the distance d in miles to the horizon when h is the height of the viewer’s eyes above the ground in feet. Round to the nearest mile. C. 2 A square garden plot has an area of 24 ft. a. Find the length of each side in simplest radical form. b. Calculate the length of each side to the nearest tenth of a foot. 3 Application Solutions: A. 4 15 r 3 3 3 3 4 (15) r 3 4 4 3 45 r3 4 11.25 3.14r 3 11.25 r 3.14 r 1.53in 3 B. d 1.5h d (1.5)(1353) d 2029.5 d 45miles C. A s 2 24 s 2 s 24 a) s 2 6 b) 4.9 ft Evaluate the following: 3 3 64 88 3 2 4 2 4 To solve: Find all factors Pull out set’s that contain the same number of terms as the root (using one number to represent the set of 4. Multiply if needed) 3 222222 4 22 4 4 4 4 4 16 44 x 4 99 4 xxxx x 3 1000 3 10 10 10 3333 4 2222 2 4 81 10 3 Evaluate the following: 3 6 6 6 3 216 6 1 1 144 2 144 12 144 2 1 (144) 2 5 32 16807 No real roots 5 25 2 5 7 7 5 486m 20n15 p3 5 (243 2)m 20n15 p3 5 (3)5 2m20n15 p3 3m 4n3 5 2 p3 Practice Problems and Answers