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Roots and powers Chapter 4

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4.1 – Estimating roots Chapter 4

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radicals Estimate each radical, and then check the real answer on your calculator. Consider whether each value is exact or an approximate.

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Pg. 206, #1–6 Independent Practice

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4.2 – irrational numbers Chapter 4

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**Rational and irrational numbers**

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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 exact value of the irrational number. approximate values exact value

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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 … which neither repeats nor terminates. c) is rational because both 8 and are perfect cubes. Its decimal form is … which is a repeating decimal.

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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

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example Use a number line to order these numbers from least to greatest.

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Pg , #4, 7, 8, 12, 15, 18, 2o Independent Practice

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**4.3 – Mixed and entire radicals**

Chapter 4

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**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

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**MIXED AND ENTIRE RADICALS**

Why? We can split a square root into its factors. The same rule applies to cube roots. Why?

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**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:

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example Simplify each radical. a) b) c)

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**example Write each radical in simplest form, if possible. a) b) c)**

Try simplifying these three:

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**example Write each mixed radical as an entire radical. a) b) c)**

Try it:

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P , #4, 5, 10 and 11(a,c,e,g,i), 14, 19, 24 Independent practice

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**4.4 – fractional exponents and radicals**

Chapter 4

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**Fill out the chart using your calculator.**

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:

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**example Evaluate each power without using a calculator. a) b) c) d)**

Try it:

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**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.

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example Evaluate: a) b) c) d)

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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.

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Pg , #3, 5, 10, 11, 12, 17, 20. Independent practice

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**4.5 – negative exponents and reciprocals**

Chapter 4

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challenge Factor: 5x2 + 41x – 36

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**Hint: try using fractions.**

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.

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**What is the rule for any number to the power of 0? Ex: 70?**

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

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**So, 73 and 7-3 are 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,

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example Evaluate each power. a) b) c) Try it:

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**example Evaluate each power without using a calculator. a) b) Recall:**

Try it (without a calculator):

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example Paleontologists use measurements from fossilized dinosaur tracks and the formula to 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.

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Pg , #3, 6, 7, 9, 13, 14, 16, 21 Independent Practice

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**4.6 – applying the exponent laws**

Chapter 4

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Exponent laws review Recall:

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Try it Find the value of this expression where a = –3 and b = 2.

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example Simplify by writing as a single power. a) b) c) d) Try these:

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example Simplify. a) b) Try this:

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challenge Simplify. There should be no negative exponents in your answer:

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example Simplify. a) b) c) d) Try these:

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**example A sphere has volume 425 m3.**

What is the radius of the sphere to the nearest tenth of a metre?

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Pg , #9, 10, 11, 12, 16, 19, 21, 22 Independent Practice

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