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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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

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 exactvalue of the irrational number.

approximate values

exact value

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

Together, the rational numbers and irrational numbers for the set of real numbers.

Real numbers

Rational numbers

Integers

Irrational numbers

Whole numbers

Natural Numbers

example

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

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

Why?

We can split a square root into its factors. The same rule applies to cube roots.

Why?

where n is a natural number, and a and b are real numbers.

We can use this rule to simplify radicals:

example

a) b) c)

example

Write each radical in simplest form, if possible.

a) b) c)

Try simplifying these three:

example

a) b) c)

Try it:

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,

Write in radical form in 2 ways.

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 husky with a body mass of 27 kg.

A polar bear with a body mass of 200 kg.

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, 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

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,

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 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.

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

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?