Roots and powers
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Roots and powers. Chapter 4. 4.1 – Estimating roots. Chapter 4. radicals. Estimate each radical, and then check the real answer on your calculator. Consider whether each value is exact or an approximate. . Pg. 206, #1–6 . Independent Practice. 4.2 – irrational numbers. Chapter 4.

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Roots and powers

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Roots and powers

Roots and powers

Chapter 4


4 1 estimating roots

4.1 – Estimating roots

Chapter 4


Radicals

radicals

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


Pg 206 1 6

Pg. 206, #1–6

Independent Practice


4 2 irrational numbers

4.2 – irrational numbers

Chapter 4


Rational and irrational numbers

Rational and irrational numbers

Rational Numbers

Irrational 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 exactvalue of the irrational number.

approximate values

exact value


Example

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 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

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


Example1

example

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


Pg 211 212 4 7 8 12 15 18 2o

Pg. 211-212, #4, 7, 8, 12, 15, 18, 2o

Independent Practice


4 3 mixed and entire radicals

4.3 – Mixed and entire radicals

Chapter 4


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

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


Mixed and entire radicals1

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

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:


Example2

example

Simplify each radical.

a) b)c)


Example3

example

Write each radical in simplest form, if possible.

a)b) c)

Try simplifying these three:


Example4

example

Write each mixed radical as an entire radical.

a) b)c)

Try it:


P 218 219 4 5 10 and 11 a c e g i 14 19 24

P. 218-219, #4, 5, 10 and 11(a,c,e,g,i), 14, 19, 24

Independent practice


4 4 fractional exponents and radicals

4.4 – fractional exponents and radicals

Chapter 4


Fractional exponents

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:


Example5

example

Evaluate each power without using a calculator.

a) b)c)d)

Try it:


Powers with rational exponents

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.


Example6

example

Evaluate:

a)b)c)d)


Example7

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.


Pg 227 228 3 5 10 11 12 17 20

Pg. 227-228, #3, 5, 10, 11, 12, 17, 20.

Independent practice


4 5 negative exponents and reciprocals

4.5 – negative exponents and reciprocals

Chapter 4


Challenge

challenge

Factor:

5x2 + 41x – 36


Consider

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

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


Reciprocals1

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,


Example8

example

Evaluate each power.

a) b) c)

Try it:


Example9

example

Evaluate each power without using a calculator.

a) b)

Recall:

Try it (without a calculator):


Example10

example

Paleontologists use measurements from fossilized dinosaur tracks and the formula to estimate the speed at

which the dinosaur travelled. In the formula, vis 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.


Pg 233 234 3 6 7 9 13 14 16 21

Pg. 233-234, #3, 6, 7, 9, 13, 14, 16, 21

Independent Practice


4 6 applying the exponent laws

4.6 – applying the exponent laws

Chapter 4


Exponent laws review

Exponent laws review

Recall:


Try it

Try it

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


Example11

example

Simplify by writing as a single power.

a)b) c) d)

Try these:


Example12

example

Simplify.

a)b)

Try this:


Challenge1

challenge

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


Example13

example

Simplify.

a) b) c)d)

Try these:


Example14

example

A sphere has volume 425 m3.

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


Pg 241 243 9 10 11 12 16 19 21 22

Pg. 241-243, #9, 10, 11, 12, 16, 19, 21, 22

Independent Practice


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