chapter 4 roots and powers. chapter 4 4.1 – estimating roots
TRANSCRIPT
Chapter 4ROOTS AND POWERS
Chapter 44.1 – ESTIMATING
ROOTS
RADICALS
Estimate each radical, and then check the real answer on your calculator. Consider whether each value is exact or an approximate.
Independent PracticePG. 206, #1–6
Chapter 44.2 – IRRATIONAL
NUMBERS
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 exact value of the irrational number.
exact value
approximate values
EXAMPLE
Tell whether each number is rational or irrational. Explain how you know.
a) b) c)
a) –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
Whole numbers
Natural Numbers
Irra
tional num
bers
EXAMPLE
Use a number line to order these numbers from least to greatest.
Independent Practice
PG. 211-212, #4, 7, 8, 12, 15, 18, 2O
Chapter 44.3 – MIXED AND
ENTIRE RADICALS
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
1 cm
2 cm
2 cm
3 cm
3 cm
4 cm
4 cm
Draw a 5 by 5 triangle. What are the two ways to write the length of the hypotenuse?
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:
Independent practice
P. 218-219, #4, 5, 10 AND 11(A,C,E,G,I), 14,
19, 24
Chapter 4
4.4 – FRACTIONAL EXPONENTS AND
RADICALS
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,
a) Write in radical form in 2 ways.
b) 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) A husky with a body mass of 27 kg.b) A polar bear with a body mass of 200 kg.
Independent practice
PG. 227-228, #3, 5, 10, 11, 12, 17, 20.
Chapter 4
4.5 – NEGATIVE EXPONENTS AND
RECIPROCALS
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, 4 and ¼ are reciprocals!
So, what is the reciprocal of ?
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.
73 = 343
What is the reciprocal of 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 atwhich the dinosaur travelled. In the formula, v is 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.
Independent Practice
PG. 233-234, #3, 6, 7, 9, 13, 14, 16, 21
Chapter 44.6 – APPLYING THE
EXPONENT LAWS
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?
Independent Practice
PG. 241-243, #9, 10, 11, 12, 16, 19, 21,
22