chapter 4 roots and powers. chapter 4 4.1 – estimating roots
TRANSCRIPT
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Chapter 4ROOTS AND POWERS
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Chapter 44.1 – ESTIMATING
ROOTS
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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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Independent PracticePG. 206, #1–6
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Chapter 44.2 – IRRATIONAL
NUMBERS
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RATIONAL AND IRRATIONAL NUMBERS
Rational Numbers
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.
exact value
approximate values
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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.
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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
Whole numbers
Natural Numbers
Irra
tional num
bers
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EXAMPLE
Use a number line to order these numbers from least to greatest.
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Independent Practice
PG. 211-212, #4, 7, 8, 12, 15, 18, 2O
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Chapter 44.3 – MIXED AND
ENTIRE RADICALS
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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
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?
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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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Independent practice
P. 218-219, #4, 5, 10 AND 11(A,C,E,G,I), 14,
19, 24
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Chapter 4
4.4 – FRACTIONAL EXPONENTS AND
RADICALS
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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,
a) Write in radical form in 2 ways.
b) 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) A husky with a body mass of 27 kg.b) A polar bear with a body mass of 200 kg.
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Independent practice
PG. 227-228, #3, 5, 10, 11, 12, 17, 20.
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Chapter 4
4.5 – NEGATIVE EXPONENTS AND
RECIPROCALS
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CHALLENGE
Factor:5x2 + 41x – 36
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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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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
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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,
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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 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.
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Independent Practice
PG. 233-234, #3, 6, 7, 9, 13, 14, 16, 21
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Chapter 44.6 – APPLYING THE
EXPONENT LAWS
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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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Independent Practice
PG. 241-243, #9, 10, 11, 12, 16, 19, 21,
22