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Roots and powers

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Chapter 4

Chapter 4

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

Independent Practice

Chapter 4

Rational 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

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.

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

Real numbers

Rational numbers

Integers

Irrational numbers

Whole numbers

Natural Numbers

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

Independent Practice

Chapter 4

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:

Simplify each radical.

a) b)c)

Write each radical in simplest form, if possible.

a)b) c)

Try simplifying these three:

Write each mixed radical as an entire radical.

a) b)c)

Try it:

Independent practice

Chapter 4

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:

Evaluate each power without using a calculator.

a) b)c)d)

Try it:

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.

Evaluate:

a)b)c)d)

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.

Independent practice

Chapter 4

Factor:

5x2 + 41x – 36

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.

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

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,

Evaluate each power.

a) b) c)

Try it:

Evaluate each power without using a calculator.

a) b)

Recall:

Try it (without a calculator):

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.

Independent Practice

Chapter 4

Recall:

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

Simplify by writing as a single power.

a)b) c) d)

Try these:

Simplify.

a)b)

Try this:

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

Simplify.

a) b) c)d)

Try these:

A sphere has volume 425 m3.

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

Independent Practice