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Foundations & Pre-Calculus 10 Chapter 4 DATE TOPIC ASSIGNMENT 4.1 Investigation and #4 4.2 Irrational Numbers 4.3 Mixed and Entire Radicals 4.4 Fractional Exponents and Radicals 4.5 Negative Exponents and Reciprocals 4.6 Applying the Exponent Laws Review Chapter 4 Test 1 Foundations & Pre-Calculus 10 4.1 Math Lab Estimating Roots Make Connections Show how you would write each as a square root, cube root, and fourth root. Square root Cube root Fourth root Show 3 as … Show 4 as … Show 5 as … Estimating Radicals Ex. #1: Estimate the following radicals to 1 decimal place. (a) 20 (b) 3 (c) 34 Check: Using your calculator. 2 4 40 Foundations & Pre-Calculus 10 Complete the following table. Radical Radicand Index Words √81 √16 16 81 √0.64 √16 √27 16 81 √0.64 √16 √27 16 81 √0.64 3 Value Exact or Approximate Foundations & Pre-Calculus 10 4.2 Irrational Numbers Investigation: Use your calculator to find the decimal representation of the following. Rational Numbers Not Rational Numbers √100 = √32 √0.25 √2 √0.24 √9 Rational Numbers • can be written as a _______________ , , where m and n are integers • have decimal representations that either ______________________ or _____________________. • Radicals that are square roots of perfect _________________, cube roots of perfect ________________ and so on Irrational Numbers • cannot be written as a _______________ , , where m and n are integers • the decimal representation neither _________________________ nor _______________________ When an irrational number is written as a radical, the radical is the value of the irrational number. We can use the square root and cube root keys on a calculator to determine the values of the irrational numbers. Ex # 1: Tell whether each number is rational or irrational. Explain how you know. a) b) √14 4 c) Foundations & Pre-Calculus 10 Together, the rational numbers and irrational numbers form the __________________________________________________ Real Numbers Ex. # 2: Use a number line to order these numbers from least to greatest. √13, √18, √9, √27, √5 5 Foundations & Pre-Calculus 10 4.3 Mixed and Entire Radicals Radicals like 18 , 3 24 , 2 , x are called ______________________. Radicals like 3 2 ,23 − 8 ,5 14 , a x are called _____________________. Ex. #1: Simplify the following. (a) √16 ! 9 (b) √16 ! √9 Multiplication Property of Radicals √#$ = % Where n is a natural number, and a and b are real numbers. We can use this property to simplify square roots and cube roots that are not perfect squares or perfect cubes, but have ___________________________ that are perfect squares or perfect cubes. To change from entire radicals to mixed radicals: 1._______________________________________________________ 2._______________________________________________________ 3. ______________________________________________________ A radical is in simplest form when the _____________________________ has no _______________________________ factors. 6 Foundations & Pre-Calculus 10 For example: Write the factors of 24 below: One factor is a perfect square. It is . Therefore we can show √24 as follows: As well, we can show √24 as follows: But we cannot simplify √24 because 24 has no factors that can be written as a fourth power. Ex #2: Simplify each radical. (a) 18 (b) (d) 48 (e) 3 50 (c) 54 (f) 7 3 16 288 Foundations & Pre-Calculus 10 To change from mixed radicals to entire radicals we: 1.____________________________________________ 2.____________________________________________ Ex #3: Write each mixed radical as an entire radical. (a) 4 3 (c) 23 5 (b) 5 11 Ex. #4: Without using a calculator arrange the following radicals from least to greatest. 7 3 ,9 2 ,5 6 , 103 ,3 17 8 Foundations & Pre-Calculus 10 4.4 Fractional Exponents and Radicals Powers with Rational Exponents When ‘m’ and ‘n’ are natural numbers, and ‘x’ is a rational number, ' % & = ' % & = and = = Another (perhaps silly) “helper” to remember where everything goes: ()* & +,,*- √& Hat goes on the top of your body, and is therefore seen in the __________________ of the exponent. Boots go on the bottom of your body and is therefore seen in the _____________________ of the exponent. So the base goes home, leaves the boots at the door and hangs up the hat. Ex. #1: Evaluate each without using a calculator. a) 27 . b) 0.49 . c) 0641 d) 9 . / 2 3 . / Foundations & Pre-Calculus 10 Ex. #2: / a) Write 40 in radical form in two ways. b) Write √3 and (√25)2 in exponent form. Ex. #3: Evaluate each. a) 0.043/2 b) 274/3 c) (-32)0.4 d) 1.81. 4 10 Foundations & Pre-Calculus 10 4.5 Negative Exponents and Reciprocals Recall: • The reciprocal of is • The reciprocal of 4 is Powers with Negative Exponents When ‘x’ is any non-zero number and ‘n’ is a rational number, & 4 is the reciprocal of & . That is, & 4 = and 5 6% = ,x≠0 Ex. #1: Evaluate each power a) 34 b) 0.3-4 c) 0 14 11 Foundations & Pre-Calculus 10 Ex. #2: Evaluate each power without using a calculator. a) 8 6/ 4 b)2 3 / **Never flip the _______________ only the _________.** Ex. #3: Paleontologists use measurements from fossilized dinosaur tracks and the : ; to formula 7 0.1558 9 estimate the speed at which the dinosaur travelled. In the formula, v is the speed in metres per second, s is the distance between successive footprints of the same foot, and f is the foot length in metres. Use the measurements in the diagram to estimate the speed of the dinosaur. 4 12 Foundations & Pre-Calculus 10 Revisiting the Exponent Laws 1. Multiplication & ⋅ & _______ # ⋅ # 0____ ! ____ ! ____ ! ____10____ ! ____1 2. Division & > & ________ $ > $ 0____ ! ____ ! ____ ! ____ ! ____ ! ____1 0____ ! ____ ! ____ ! ____1 3. Power of a Power 0& 1 ________ 0? 1 0____ ! ____1 0____ ! ____10____ ! ____10____ ! ____1 4. Power of a Product 0&@1 ___________ 0A91 0______10______10______1 ____ ! ____ ! ____ ! ____ ! ____ ! ____ 5 5. Power of a Quotient 2 3 B C 2D 3 23 23 23 = 6. Zero Exponent: & E ______ Note the difference between the following: 1. 041 2. 4 13 Foundations & Pre-Calculus 10 From Math 9 Chapter 2 Review: More Practice 1. 0& @ 10& @ 1 2. ) F; 3. 05 1 4. 0# $ 1 0#$ 1 ) F 5. 051/ 14 05 % B 1/ 05 % B 1 Foundations & Pre-Calculus 10 4.6 Applying the Exponent Laws Recall the exponent laws for integer bases and whole number exponents. Product of Powers: Power of a product: Quotient of Powers: Power of a quotient: Power of a power: ) ; FG What is the value of 2 H 3 ) F 4 when # 3 and $ 2? We can use the exponent laws to _________________________ expressions that contain rational number bases. It is a convention to write a simplified power with a exponent. 15 Foundations & Pre-Calculus 10 Ex. #1: Simplify by writing as a single power. Explain the reasoning. (Don’t write too big) 16 Foundations & Pre-Calculus 10 Ex. #2: Simplify. Explain the reasoning. Ex. #3: Simplify. Explain the reasoning. 17 Foundations & Pre-Calculus 10 Ex. #4: A sphere has a volume of 425 m3. What is the radius of the sphere to the nearest tenth of a metre? 18