7 Irrational numbersAn 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 valuesexact value
8 exampleTell 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.
9 The number systemTogether, the rational numbers and irrational numbers for the set of real numbers.Real numbersRational numbersIntegersIrrational numbersWhole numbersNatural Numbers
10 exampleUse a number line to order these numbers from least to greatest.
13 Mixed and entire radicals Draw the following triangles on the graph paper that has been distributed, and label the sides of the hypotenuses.1 cm4 cm1 cm3 cm3 cm2 cm4 cmDraw a 5 by 5 triangle. What are the two ways to write the length of the hypotenuse?2 cm
14 MIXED AND ENTIRE RADICALS Why?We can split a square root into its factors. The same rule applies to cube roots.Why?
15 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:
17 example Write each radical in simplest form, if possible. a) b) c) Try simplifying these three:
18 example Write each mixed radical as an entire radical. a) b) c) Try it:
19 P , #4, 5, 10 and 11(a,c,e,g,i), 14, 19, 24Independent practice
20 4.4 – fractional exponents and radicals Chapter 4
21 Fill out the chart using your calculator. Fractional exponentsFill 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:
22 example Evaluate each power without using a calculator. a) b) c) d) Try it:
23 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.
25 exampleBiologists 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.
29 Hint: try using fractions. considerThis 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.
30 What is the rule for any number to the power of 0? Ex: 70? reciprocalsTwo 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
31 So, 73 and 7-3 are reciprocals. 73 ・ 7-3 = 70So, 73 and 7-3 are reciprocals.What is the reciprocal of 343?73 = 343When x is any non-zero number and n is a rational number, x-n is the reciprocal of xn. That is,
33 example Evaluate each power without using a calculator. a) b) Recall: Try it (without a calculator):
34 examplePaleontologists use measurements from fossilized dinosaur tracks and the formula to estimate the speed atwhich 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.