1. Roots and powers
Chapter 4

2. 4.1 – Estimating roots
Chapter 4

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

4. Pg. 206, #1–6
Independent Practice

5. 4.2 – irrational numbers
Chapter 4

6. Rational and irrational numbers

7. 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

8. 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.

9. 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

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

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

12. 4.3 – Mixed and entire radicals
Chapter 4

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 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

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:

16. example
Simplify each radical.
a) b) c)

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, 24
Independent practice

20. 4.4 – fractional exponents and radicals
Chapter 4

21. 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:

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.

24. example
Evaluate:
a) b) c) d)

25. 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.

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

27. 4.5 – negative exponents and reciprocals
Chapter 4

28. challenge
Factor:
5x2 + 41x – 36

29. 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.

30. 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

31. 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,

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

33. example Evaluate each power without using a calculator. a) b) Recall:
Try it (without a calculator):

34. 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.

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

36. 4.6 – applying the exponent laws
Chapter 4

37. Exponent laws review
Recall:

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

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

40. example
Simplify.
a) b)
Try this:

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

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

43. example A sphere has volume 425 m3.
What is the radius of the sphere to the nearest tenth of a metre?

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