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## Roots and powers

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**Roots and powers**Chapter 4**4.1 – Estimating roots**Chapter 4**radicals**Estimate each radical, and then check the real answer on your calculator. Consider whether each value is exact or an approximate.**Pg. 206, #1–6**Independent Practice**4.2 – irrational numbers**Chapter 4**Rational and irrational numbers**Rational Numbers Irrational Numbers**Irrational numbers**An irrational number cannot be written in the form m/n, where m and n are integers and n ≠ 0. The decimal representation of an irrational number neither terminates nor repeats. When an irrational number is written as a radical, the radical is the exactvalue of the irrational number. approximate values exact value**example**Tell whether each number is rational or irrational. Explain how you know. a) b) c) –3/5 is rational, because it’s written as a fraction. In its decimal form it’s –0.6, which terminates. b) is irrational since 14 is not a perfect square. The decimal form is 3.741657387… which neither repeats nor terminates. c) is rational because both 8 and 27 are perfect cubes. Its decimal form is 0.6666666… which is a repeating decimal.**The number system**Together, the rational numbers and irrational numbers for the set of real numbers. Real numbers Rational numbers Integers Irrational numbers Whole numbers Natural Numbers**example**Use a number line to order these numbers from least to greatest.**Pg. 211-212, #4, 7, 8, 12, 15, 18, 2o**Independent Practice**4.3 – Mixed and entire radicals**Chapter 4**Mixed and entire radicals**Draw the following triangles on the graph paper that has been distributed, and label the sides of the hypotenuses. 1 cm 4 cm 1 cm 3 cm 3 cm 2 cm 4 cm Draw a 5 by 5 triangle. What are the two ways to write the length of the hypotenuse? 2 cm**MIXED AND ENTIRE RADICALS**Why? We can split a square root into its factors. The same rule applies to cube roots. Why?**Multiplication properties of radicals**where n is a natural number, and a and b are real numbers. We can use this rule to simplify radicals:**example**Simplify each radical. a) b) c)**example**Write each radical in simplest form, if possible. a) b) c) Try simplifying these three:**example**Write each mixed radical as an entire radical. a) b) c) Try it:**P. 218-219, #4, 5, 10 and 11(a,c,e,g,i), 14, 19, 24**Independent practice**4.4 – fractional exponents and radicals**Chapter 4**Fractional exponents**Fill out the chart using your calculator. What do you think it means when a power has an exponent of ½? What do you think it means when a power has an exponent of 1/3? Recall the exponent law: When n is a natural number and x is a rational number:**example**Evaluate each power without using a calculator. a) b) c) d) Try it:**Powers with rational exponents**When m and n are natural numbers, and x is a rational number, Write in radical form in 2 ways. Write and in exponent form.**example**Evaluate: a) b) c) d)**example**Biologists use the formula b = 0.01m2/3 to estimate the brain mass, b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass of each animal. A husky with a body mass of 27 kg. A polar bear with a body mass of 200 kg.**Pg. 227-228, #3, 5, 10, 11, 12, 17, 20.**Independent practice**4.5 – negative exponents and reciprocals**Chapter 4**challenge**Factor: 5x2 + 41x – 36**consider**This rectangle has an area of 1 square foot. List 5 possible pairs of lengths and widths for this rectangle. (Remember, they will need to have a product of 1). Hint: try using fractions.**reciprocals**Two numbers with a product of 1 are reciprocals. So, what is the reciprocal of ? So, 4 and ¼ are reciprocals! What is the rule for any number to the power of 0? Ex: 70? If we have two powers with the same base, and their exponents add up to 0, then they must be reciprocals. Ex: 73 ・ 7-3 = 70**Reciprocals**73 ・ 7-3 = 70 So, 73 and 7-3 are reciprocals. What is the reciprocal of 343? 73 = 343 When x is any non-zero number and n is a rational number, x-n is the reciprocal of xn. That is,**example**Evaluate each power. a) b) c) Try it:**example**Evaluate each power without using a calculator. a) b) Recall: Try it (without a calculator):**example**Paleontologists use measurements from fossilized dinosaur tracks and the formula to estimate the speed at which the dinosaur travelled. In the formula, vis the speed inmetres per second, s is the distance between successivefootprints of the same foot, and f is the foot length in metres.Use the measurements in the diagram to estimate the speed ofthe dinosaur.**Pg. 233-234, #3, 6, 7, 9, 13, 14, 16, 21**Independent Practice**4.6 – applying the exponent laws**Chapter 4**Exponent laws review**Recall:**Try it**Find the value of this expression where a = –3 and b = 2.**example**Simplify by writing as a single power. a) b) c) d) Try these:**example**Simplify. a) b) Try this:**challenge**Simplify. There should be no negative exponents in your answer:**example**Simplify. a) b) c) d) Try these:**example**A sphere has volume 425 m3. What is the radius of the sphere to the nearest tenth of a metre?**Pg. 241-243, #9, 10, 11, 12, 16, 19, 21, 22**Independent Practice