© boardworks ltd 2008 1 of 70 n2 powers, roots and standard form maths age 14-16

© Boardworks Ltd 2008 1 of 70

N2 Powers, roots and standard form

Maths Age 14-16


A

A

A

A

A

A

N2.1 Powers and roots

Contents

N2.2 Index laws

N2.4 Fractional indices

N2.5 Surds


N2.3 Negative indices and reciprocals

N2.6 Standard form


Square numbers

When we multiply a number by itself we say that we are squaring the number.

To square a number we can write a small 2 after it.

For example, the number 3 multiplied by itself can be written as

3 × 3 or 32

The value of three squared is 9.

The result of any whole number multiplied by itself is called a square number.


Finding the square root is the inverse of finding the square:

8

squared

64

square rooted

We write

64 = 8

The square root of 64 is 8.

Square roots


The product of two square numbers

The product of two square numbers is always another square number.

For example,

4 × 25 = 100

2 × 2 × 5 × 5 = 2 × 5 × 2 × 5

and

(2 × 5)2 = 102

We can use this fact to help us find the square roots of larger square numbers.

because


Using factors to find square roots

If a number has factors that are square numbers then we can use these factors to find the square root.

For example,

= 2 × 10

= 20

√400 = √(4 × 100)

Find 400

= 3 × 5

= 15

√225 = √(9 × 25)

Find 225

= √4 × √100 = √9 × √25


Finding square roots of decimals

We can also find the square root of a number can be made be dividing two square numbers.

For example,

= 3 ÷ 10

= 0.3

0.09 = (9 ÷ 100)

Find 0.09

= 12 ÷ 100

= 0.12

0.0144 = (144 ÷ 10000)

Find 0.0144

= √9 ÷ √100 = √144 ÷ √10000


If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly.

Use the key on your calculator to find out 2.

The calculator shows this as 1.414213562

This is an approximation to 9 decimal places.

The number of digits after the decimal point is infinite and non-repeating.

Approximate square roots

This is an example of an irrational number.


Estimating square roots

What is 50?

50 is not a square number but lies between 49 and 64.

Therefore,

49 < 50 < 64

So,

7 < 50 < 8

Use the key on you calculator to work out the answer.

50 = 7.07 (to 2 decimal places.)

50 is much closer to 49 than to 64, so 50 will

be about 7.1


Negative square roots

5 × 5 = 25 and –5 × –5 = 25

Therefore, the square root of 25 is 5 or –5.

When we use the symbol we usually mean the positive square root.

However the equation,

x2 = 25

has 2 solutions,

x = 5 or x = –5

We can also write ± to mean both the positive and the negative square root.


Squares and square roots from a graph


The numbers 1, 8, 27, 64, and 125 are all:

Cube numbersCube numbers

13 = 1 × 1 × 1 = 1 ‘1 cubed’ or ‘1 to the power of 3’





Cubes


Cube roots

Finding the cube root is the inverse of finding the cube:

5

cubed

125

cube rooted

We write

The cube root of 125 is 5.

125 = 53


Squares, cubes and roots


Index notation

We use index notation to show repeated multiplication by the same number.

For example:

we can use index notation to write 2 × 2 × 2 × 2 × 2 as

25

This number is read as ‘two to the power of five’.

25 = 2 × 2 × 2 × 2 × 2 = 32

base

Index or power


Evaluate the following:

62 = 6 × 6 = 36

34 = 3 × 3 × 3 × 3 = 81

(–5)3 = –5 × –5 × –5 = –125

27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

(–1)5 = –1 × –1 × –1 × –1 × –1 = –1

(–4)4 = –4 × –4 × –4 × –4 = 256

When we raise a negative number to an odd power the

answer is negative.

When we raise a negative number to an even power the answer is positive.

Index notation


Using a calculator to find powers

We can use the xy key on a calculator to find powers.

For example:

to calculate the value of 74 we key in:

The calculator shows this as 2401.

74 = 7 × 7 × 7 × 7 = 2401

7 xy 4 =


A

A

A

A

A

A

N2.2 Index laws

Contents


N2.5 Surds



N2.6 Standard form



Multiplying numbers in index form

When we multiply two numbers written in index form and with the same base we can see an interesting result.

For example:

34 × 32 = (3 × 3 × 3 × 3) × (3 × 3)= 3 × 3 × 3 × 3 × 3 × 3= 36

73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7)= 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7= 78

What do you notice?When we multiply two numbers with the same base the

indices are added. In general, xm × xn = x(m + n)

When we multiply two numbers with the same base the indices are added. In general, xm × xn = x(m + n)

= 3(4 + 2)

= 7(3 + 5)


Dividing numbers in index form

When we divide two numbers written in index form and with the same base we can see another interesting result.

For example:

45 ÷ 42 =4 × 4 × 4 × 4 × 4

4 × 4= 4 × 4 × 4 = 43

56 ÷ 54 =5 × 5 × 5 × 5 × 5 × 5

5 × 5 × 5 × 5= 5 × 5 = 52

What do you notice?

= 4(5 – 2)

= 5(6 – 4)

When we divide two numbers with the same base the indices are subtracted. In general, xm ÷ xn = x(m – n)

When we divide two numbers with the same base the indices are subtracted. In general, xm ÷ xn = x(m – n)


What do you notice?

Raising a power to a power

Sometimes numbers can be raised to a power and the result raised to another power.For example,

(43)2 = 43 × 43

= (4 × 4 × 4) × (4 × 4 × 4)

= 46

When a number is raised to a power and then raised to another power, the powers are multiplied. In general, (xm)n = xmn

When a number is raised to a power and then raised to another power, the powers are multiplied. In general, (xm)n = xmn

= 4(3 × 2)


Using index laws


The power of 1

Any number raised to the power of 1 is equal to the number itself. In general, x1 = x

Any number raised to the power of 1 is equal to the number itself. In general, x1 = x

Find the value of the following using your calculator:

61 471 0.91 –51 01

Because of this we don’t usually write the power when a number is raised to the power of 1.


The power of 0

Look at the following division:

64 ÷ 64 = 1

Using the second index law,

64 ÷ 64 = 6(4 – 4) = 60

That means that: 60 = 1

For example,

100 = 1 3.4520 = 1 723 538 5920 = 1

Any non-zero number raised to the power of 0 is equal to 1.Any non-zero number raised to the power of 0 is equal to 1.


x0 = 1 (for x = 0)x0 = 1 (for x = 0)

Here is a summery of the index laws you have met so far:

xm × xn = x(m + n)

xm ÷ xn = x(m – n)xm ÷ xn = x(m – n)

Index laws

(xm)n = xmn(xm)n = xmn

x1 = xx1 = x


A

A

A

A

A

A


Contents


N2.5 Surds

N2.2 Index laws


N2.6 Standard form



Negative indices

Look at the following division:

32 ÷ 34 =3 × 3

3 × 3 × 3 × 3=

13 × 3

=132

Using the second index law,

32 ÷ 34 = 3(2 – 4) = 3–2

That means that 3–2 = 132

Similarly, 6–1 = 16

7–4 = 174

and 5–3 = 153


Reciprocals

A number raised to the power of –1 gives us the reciprocal of that number.

The reciprocal of a number is what we multiply the number by to get 1.

The reciprocal of a is 1

a

The reciprocal of is a

b

b

a

We can find reciprocals on a calculator using the key.x-1


Finding the reciprocals

Find the reciprocals of the following:

1) 6 The reciprocal of 6 = 1

66-1 = 1

6or

2)3

7The reciprocal of =

7

3

3

7or

3

7=

7

3

–1

3) 0.8 0.8 =4

5The reciprocal of =

5

4

4

5= 1.25

or 0.8–1 = 1.25


Match the reciprocal pairs


Here is a summery of the index laws for negative indices.

Index laws for negative indices

x–1 = 1x

x–n = 1xn

The reciprocal of x is 1

x

The reciprocal of xn is 1

xn


A

A

A

A

A

A


Contents

N2.5 Surds

N2.2 Index laws



N2.6 Standard form



Indices can also be fractional. Suppose we have

Fractional indices

9 × 9 =12

12 9 + =

12

12 91 = 9

But, 9 × 9 = 9

In general, x = x x = x 12

Similarly, 81 = 8

But,

In general,

8 × 8 × 8 =13

13

13 8 + + =

13

13

13

8 × 8 × 8 = 83 3 3

x = x x = x 13 3

Because 3 × 3 = 9

9 .12

Because 2 × 2 × 2 = 8


Fractional indices

Using the rule that (xa)b = xab we can write

In general,

x = (x)mx = (x)mmn n

What is the value of 25 ?32

We can think of as 25 .2532

12 × 3

1225 × 3 = (25)3

= (5)3

= 125


Evaluate the following

1) 49 12

2) 100023

3) 8- 13

4) 64- 23

5) 452

√49 =49 12 = 7

(3√1000)2 =1000 23 = 102 = 100

8- 13 =

18

13

=1

3√8=

12

64- 23 =

164

23

=1

(3√64)2=

142

=116

(√4)5 =4 52 = 25 = 32


Here is a summery of the index laws for fractional indices.

Index laws for fractional indices

x = x 1n

n

x = x 12

x = xm or (x)mnmn n


A

A

A

A

A

A

N2.5 Surds

Contents


N2.2 Index laws



N2.6 Standard form



Surds

The square roots of many numbers cannot be found exactly.

For example, the value of √3 cannot be written exactly as a fraction or a decimal.

For this reason it is often better to leave the square root sign in and write the number as √3.

Which one of the following is not a surd?√2, √6 , √9 or √14

√3 is an example of a surd.

The value of √3 is an irrational number.

9 is not a surd because it can be written exactly.


Multiplying surds

What is the value of √3 × √3?

We can think of this as squaring the square root of three.

Squaring and square rooting are inverse operations so,

√3 × √3 = 3

In general, √a × √a = aIn general, √a × √a = a

What is the value of √3 × √3 × √3?

Using the above result,

√3 × √3 × √3 = 3 × √3

= 3√3

Like algebra, we do not use the × sign

when writing surds.


Multiplying surds

Use a calculator to find the value of √2 × √8.

What do you notice?

√2 × √8 = 4

4 is the square root of 16 and 2 × 8 = 16.

Use a calculator to find the value of √3 × √12.

√3 × √12 = 6

6 is the square root of 36 and 3 × 12 = 36.

In general, √a × √b = √ab In general, √a × √b = √ab

(= √16)

(= √36)


Dividing surds

Use a calculator to find the value of √20 ÷ √5.

What do you notice?

√20 ÷ √5 = 2

2 is the square root of 4 and 20 ÷ 5 = 4.

Use a calculator to find the value of √18 ÷ √2.

√18 ÷ √2 = 3

3 is the square root of 9 and 18 ÷ 2 = 9.

In general, √a ÷ √b =ab

(= √4)

(= √9)


Simplifying surds

We are often required to simplify surds by writing them in the form a√b. For example,

Simplify √50 by writing it in the form a√b.

Start by finding the largest square number that divides into 50.

This is 25. We can use this to write:

√50 = √(25 × 2)

= √25 × √2

= 5√2


Simplifying surds

Simplify the following surds by writing them in the form a√b.

1) √45 2) √24 3) √300

√45 = √(9 × 5)

= √9 × √5

= 3√5

√24 = √(4 × 6)

= √4 × √6

= 2√6

√300 = √(100 × 3)

= √100 × √3

= 10√3


Simplifying surds


Adding and subtracting surds

Surds can be added or subtracted if the number under the square root sign is the same. For example,

Simplify √27 + √75.

Start by writing √27 and √75 in their simplest forms.

√27 = √(9 × 3)

= √9 × √3

= 3√3

√75 = √(25 × 3)

= √25 × √3

= 5√3

√27 + √75 = 3√3 + 5√3 = 8√3


Perimeter and area problem

The following rectangle has been drawn on a square grid.

Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form.

1

3 √10

6

22√10

Width = √(32 + 12)

= √(9 + 1)

= √10 units

Length = √(62 + 22)

= √(36 + 4)

= √40

= 2√10 units


Perimeter and area problem

The following rectangle has been drawn on a square grid.

Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form.

1

3 √10

6

22√10

Perimeter = √10 + 2√10 +

= 6√10 units

√10 + 2√10

Area = √10 × 2√10

= 2 × √10 × √10

= 2 × 10

= 20 units2


Rationalizing the denominator

When a fraction has a surd as a denominator we usually rewrite it so that the denominator is a rational number.

This is called rationalizing the denominator.

Simplify the fraction5√2

Remember, if we multiply the numerator and the denominator of a fraction by the same number the value of the fraction remains unchanged.

In this example, we can multiply the numerator and the denominator by √2 to make the denominator into a whole number.



When a fraction has a surd as a denominator we usually rewrite it so that the denominator is a rational number.

This is called rationalizing the denominator.

Simplify the fraction5

√2

5√2

=

×√2

×√2

25√2



2√3

=

×√3

×√3

32√3

Simplify the following fractions by rationalizing their denominators.

1)2√3

2)√2√5

3)3

4√7

√2√5

=

×√5

×√5

5√10 3

4√7=

×√7

×√7

283√7


A

A

A

A

A

AN2.6 Standard form

Contents

N2.5 Surds


N2.2 Index laws





Powers of ten

Our decimal number system is based on powers of ten.

We can write powers of ten using index notation.

10 = 101

100 = 10 × 10 = 102

1000 = 10 × 10 × 10 = 103

10 000 = 10 × 10 × 10 × 10 = 104

100 000 = 10 × 10 × 10 × 10 × 10 = 105

1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106 …


Negative powers of ten

Any number raised to the power of 0 is 1, so

1 = 100

Decimals can be written using negative powers of ten

0.01 = = = 10-21102

1100

0.001 = = = 10-31103

11000

0.0001 = = = 10-4110000

1104

0.00001 = = = 10-51100000

1105

0.000001 = = = 10-6 …11000000

1106

0.1 = = =10-1110

1101


Very large numbers

Use you calculator to work out the answer to 40 000 000 × 50 000 000.

Your calculator may display the answer as:

What does the 15 mean?

The 15 means that the answer is 2 followed by 15 zeros or:

2 × 10152 × 1015 = 2 000 000 000 000 000

2 E15 or 2 152 ×10

15 ,


Very small numbers

Use you calculator to work out the answer to 0.0002 ÷ 30 000 000.

Your calculator may display the answer as:

What does the –12 mean?

The –12 means that the 15 is divided by 1 followed by 12 zeros.

1.5 × 10-121.5 × 10-12 = 0.000000000002

1.5 E–12 or 1.5 –121.5 ×10

–12 ,


Standard form

2 × 1015 and 1.5 × 10-12 are examples of a number written in standard form.

Numbers written in standard form have two parts:

A number between 1

and 10

A number between 1

and 10× A power of

10A power of

10

This way of writing a number is also called standard index form or scientific notation.

Any number can be written using standard form, however it is usually used to write very large or very small numbers.


Standard form – writing large numbers

For example, the mass of the planet earth is about 5 970 000 000 000 000 000 000 000 kg.

We can write this in standard form as a number between 1 and 10 multiplied by a power of 10.

5.97 × 1024 kg5.97 × 1024 kg

A number between 1 and 10

A power of ten


How can we write these numbers in standard form?

80 000 000 = 8 × 107

230 000 000 = 2.3 × 108

724 000 = 7.24 × 105

6 003 000 000 = 6.003 × 109

371.45 = 3.7145 × 102



These numbers are written in standard form. How can they be written as ordinary numbers?

5 × 1010 = 50 000 000 000

7.1 × 106 = 7 100 000

4.208 × 1011 = 420 800 000 000

2.168 × 107 = 21 680 000

6.7645 × 103 = 6764.5



We can write very small numbers using negative powers of ten.

We write this in standard form as:

For example, the width of this shelled amoeba is 0.00013 m.

A number between 1 and 10

A negative power of 10

Standard form – writing small numbers

1.3 × 10-4 m.1.3 × 10-4 m.


How can we write these numbers in standard form?

0.0006 = 6 × 10-4

0.00000072 = 7.2 × 10-7

0.0000502 = 5.02 × 10-5

0.0000000329 = 3.29 × 10-8

0.001008 = 1.008 × 10-3



8 × 10-4 = 0.0008

2.6 × 10-6 = 0.0000026

9.108 × 10-8 = 0.00000009108

7.329 × 10-5 = 0.00007329

8.4542 × 10-2 = 0.084542


These numbers are written in standard form. How can they be written as ordinary numbers?


Which number is incorrect?


Ordering numbers in standard form

Write these numbers in order from smallest to largest:5.3 × 10-4, 6.8 × 10-5, 4.7 × 10-3, 1.5 × 10-4.

To order numbers that are written in standard form start by comparing the powers of 10.

Remember, 10-5 is smaller than 10-4. That means that 6.8 × 10-

5 is the smallest number in the list.

When two or more numbers have the same power of ten we can compare the number parts. 5.3 × 10-4 is larger than 1.5 × 10-4 so the correct order is:

6.8 × 10-5, 1.5 × 10-4, 5.3 × 10-4, 4.7 × 10-3


Ordering planet sizes


Calculations involving standard form

What is 2 × 105 multiplied by 7.2 × 103 ?

To multiply these numbers together we can multiply the number parts together and then the powers of ten together.

2 × 105 × 7.2 × 103 = (2 × 7.2) × (105 × 103)

= 14.4 × 108

This answer is not in standard form and must be converted!

14.4 × 108 = 1.44 × 10 × 108

= 1.44 × 109



What is 1.2 × 10-6 divided by 4.8 × 107 ?

To divide these numbers we can divide the number parts and then divide the powers of ten.

(1.2 × 10-6) ÷ (4.8 × 107) = (1.2 ÷ 4.8) × (10-6 ÷ 107)

= 0.25 × 10-13

This answer is not in standard form and must be converted.

0.25 × 10-13 = 2.5 × 10-1 × 10-13

= 2.5 × 10-14


Travelling to Mars

How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?



Time to reach Mars =8.32 × 107

2.6 × 103

= 3.2 × 104 hours

Rearrange speed =distance

timetime =

distancespeed

to give

This is 8.32 ÷ 2.6

This is 107 ÷ 103

How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?



Use your calculator to work out how long 3.2 × 104 hours is in years.

You can enter 3.2 × 104 into your calculator using the EXP key:

3 . 2 EXP 4

Divide by 24 to give the equivalent number of days.

Divide by 365 to give the equivalent number of years.

3.2 × 104 hours is over 3½ years.

© boardworks ltd 2008 1 of 70 n2 powers, roots and standard form maths age 14-16

Documents