4 ESO Academics - UNIT 02 - POWERS, ROOTS AND LOGARITHMS

Unit 02 October

1. INTEGER EXPONENT POWERS.

1.1. INDEX OR EXPONENT NOTATION.

Instead of writing 2 𝑥𝑥 2 𝑥𝑥 2 𝑥𝑥 2 𝑥𝑥 2 we can write 25:

In 25, the 2 is called the base number and the 5 is the index, power or

exponent. The index is the number of times the base number appears in the product.

This notation enables us to quickly write long lists of identical numbers being

multiplied together.

34 is the short way of writing 3 𝑥𝑥 3 𝑥𝑥 3 𝑥𝑥 3

106 is the short way of writing 1,000,000 = 10𝑥𝑥10𝑥𝑥10𝑥𝑥10𝑥𝑥10𝑥𝑥10

MATH VOCABULARY: Base Number, Index, Exponent. Distribution.

1.2. NAMING POWERS.

65 may be read as:

• Six to the fifth power

• Six to the power of five

• Six powered to five.

NOTE: During this curse use the first way.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.1

Unit 02 October

1.3. SQUARE AND CUBE POWERS.

We call Square Power a number to the second power. It represent the result of

multiplying a number by itself. The verb "to square" is used to denote this operation.

Squaring is the same as raising to the power 2.

We call Cube Power a number to the third power. It represent the result of the

number multiplied by itself twice. The verb "to cube" is used to denote this operation.

Cubing is the same as raising to the power 3.

MATH VOCABULARY: Square, Cube, Raise, To Factorize.


Unit 02 October

1.4. INTEGER EXPONENT POWERS.

If 𝒂𝒂 ∈ and ℝ 𝐧𝐧 ∈ ℕ ⇒

𝐚𝐚𝐧𝐧 = 𝐚𝐚 ⋅ … ⋅ 𝐚𝐚��𝐧𝐧 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯

𝒂𝒂−𝒏𝒏 =𝟏𝟏𝒂𝒂𝒏𝒏 =

𝟏𝟏𝐚𝐚 ⋅ … ⋅ 𝐚𝐚��𝐧𝐧 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯

;𝒂𝒂 ≠ 𝟎𝟎

Moreover, ∀𝐚𝐚 ≠ 𝟎𝟎;𝐛𝐛 ≠ 𝟎𝟎 ⇒

𝐚𝐚𝟎𝟎 = 𝟏𝟏 𝐚𝐚𝟏𝟏 = 𝐚𝐚 𝐚𝐚−𝟏𝟏 =𝟏𝟏𝐚𝐚 �

𝐚𝐚𝐛𝐛�

−𝟏𝟏=𝐛𝐛𝐚𝐚

23 = 2 ∙ 2 ∙ 2 = 8

2−3 =1

2 ∙ 2 ∙ 2 =18

(−5)−3 =1

(−5) ∙ (−5) ∙ (−5) = −1

125

�2−3�

−1

=−32 = −

32

1.5. PROPERTIES OF POWERS.

When powers with the same base are multiplied, the base remains unchanged

and the exponents are added.

𝒂𝒂𝒏𝒏 ∙ 𝒂𝒂𝒎𝒎 = 𝒂𝒂𝒏𝒏+𝒎𝒎

75 ∙ 73 = (7 ∙ 7 ∙ 7 ∙ 7 ∙ 7) ∙ (7 ∙ 7 ∙ 7) = 75+3 = 78


Unit 02 October

When we have powers with different base but the same exponent, we multiply

the bases and keep the same exponent. We may also do it in the reciprocal way.

(𝒂𝒂 ∙ 𝒃𝒃)𝒏𝒏 = 𝒂𝒂𝒏𝒏 ∙ 𝒃𝒃𝒏𝒏

53 ∙ 73 = (5 ∙ 5 ∙ 5) ∙ (7 ∙ 7 ∙ 7) = (5 ∙ 7) ∙ (5 ∙ 7) ∙ (5 ∙ 7) = (5 ∙ 7)3 = 353

64 = (3 ∙ 2)4 = 34 ∙ 24

When powers with the same base are divided, the base remains unchanged

and the exponents are subtracted.

𝒂𝒂𝒏𝒏 ÷ 𝒂𝒂𝒎𝒎 =𝒂𝒂𝒏𝒏

𝒂𝒂𝒎𝒎 = 𝒂𝒂𝒏𝒏−𝒎𝒎

75 ÷ 73 = (7 ∙ 7 ∙ 7 ∙ 7 ∙ 7) ÷ (7 ∙ 7 ∙ 7) =7 ∙ 7 ∙ 7 ∙ 7 ∙ 7

7 ∙ 7 ∙ 7 = 75−3 = 72

When we have powers with different base but the same exponent, we divide

the bases and keep the same exponent. We may also do it in the reciprocal way.

(𝒂𝒂 ÷ 𝒃𝒃)𝒏𝒏 = �𝒂𝒂𝒃𝒃�

𝒏𝒏=𝒂𝒂𝒏𝒏

𝒃𝒃𝒏𝒏 = 𝒂𝒂𝒏𝒏 ÷ 𝒃𝒃𝒏𝒏

153 ÷ 33 = (15 ∙ 15 ∙ 15) ÷ (3 ∙ 3 ∙ 3) = (15 ÷ 3) ∙ (15 ÷ 3) ∙ (15 ÷ 3) = (15 ÷ 3)3

= �153 �

3

= 53

When we have a power of powers, the exponents must be multiplied:

(𝒂𝒂𝒏𝒏)𝒎𝒎 = 𝒂𝒂𝒏𝒏∙𝒎𝒎

(23)5 = 23 ∙ 23 ∙ 23 ∙ 23 ∙ 23 = 23+3+3+3+3 = 23∙5 = 215


Unit 02 October

2. ROOTS AND RADICALS.

The nth root of a number is written as 𝐚𝐚 √𝐚𝐚, called𝐧𝐧 radical, and is the number

that must be multiplied by itself n times to equal the number . 𝐚𝐚

√7293 = 9 ⇔ 93 = 729

√−3433 = −7 ⇔ (−7)3 = −343

We have different types of radicals:

√𝒂𝒂𝒏𝒏

Radicand Index Number of Roots

𝒂𝒂 > 𝟎𝟎 n odd 1 root: positive

n even 2 roots: 1 positive and its opposite

𝒂𝒂 = 𝟎𝟎 n odd or even 1 root: √𝟎𝟎𝒏𝒏 = 𝟎𝟎

𝒂𝒂 < 𝟎𝟎 n odd 1 root: negative

n even no Real root

MATH VOCABULARY: Root, Radical, Radicand, Index.


Unit 02 October

3. FRACTIONAL EXPONENTS.

Radical expressions can be rewritten using fractional exponents, so radicals can

be expressed as powers.

√𝐚𝐚𝐧𝐧 = 𝐚𝐚𝟏𝟏𝐧𝐧 ⇔ �𝐚𝐚

𝟏𝟏𝐧𝐧�

𝐧𝐧= 𝐚𝐚

𝐧𝐧𝐧𝐧 = 𝐚𝐚𝟏𝟏 = 𝐚𝐚

√𝒂𝒂𝒎𝒎𝐧𝐧 = 𝐚𝐚𝒎𝒎𝐧𝐧 ⇔ �𝐚𝐚

𝒎𝒎𝐧𝐧 �

𝐧𝐧= 𝐚𝐚

𝐦𝐦∙𝐧𝐧𝐧𝐧 = 𝐚𝐚𝐦𝐦

�√7296 �2

= ��936 �2

= �936�

2= �9

12�

2= 9

22 = 91 = 9

Two radicals are equivalents when both can be expressed as fractional

exponent powers, with the same base and equivalent index.

𝐚𝐚𝐦𝐦𝐧𝐧 𝐢𝐢𝐯𝐯 𝐯𝐯𝐞𝐞𝐞𝐞𝐢𝐢𝐯𝐯𝐚𝐚𝐞𝐞𝐯𝐯𝐧𝐧𝐭𝐭 𝐭𝐭𝐭𝐭 𝐛𝐛

𝐩𝐩𝐞𝐞 ⇔ 𝐚𝐚 = 𝐛𝐛 𝐚𝐚𝐧𝐧𝐚𝐚

𝐦𝐦𝐧𝐧 =

𝐩𝐩𝐞𝐞

254 𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑡𝑡 2

108

Since we can represent radicals as powers, all the properties of powers are

applied to radicals.

MATH VOCABULARY: Equivalent Radicals.


Unit 02 October

4. OPERATION WITH RADICALS.

4.1. REDUCTION OF RADICALS TO SAME INDEX.

To reduce radicals to the same index we have to express them as fractional

exponent powers and look for the same denominator using the lowest common

multiple (LCM).

Reduce to the same index: √5, √743 :

√5 = 512; �743 = 7

43

We have to do the LCM of the exponent:

𝐿𝐿𝐿𝐿𝐿𝐿(2,3) = 6 ⇒12 =

36 𝑒𝑒𝑒𝑒𝑎𝑎

43 =

86

Therefore:

512 = 5

36 = �536 𝑒𝑒𝑒𝑒𝑎𝑎 7

34 = 7

86 = �786

MATH VOCABULARY: Lowest Common Multiple (LCM), Highest Common Factor (HCF).

4.2. SIMPLIFYING RADICALS.

You can simplify radicals by expressing them as fractional exponent powers and

obtaining the simplest fraction of the exponent.

Simplify √3525

�3525 = 3525 = 3

15 = √35


Unit 02 October

MATH VOCABULARY: Simplest Fraction.

4.3. EXTRACTING FACTORS.

To extract factors from a radical, we have to factorize the radicand and express

its factors as powers. Then we have to extract the factors which exponent is higher or

equal than the radical index by using the powers properties.

Example 1:

�3,8883

3,888 = 24 ∙ 35 ⇒ �3,8883 = �24 ∙ 353

The factor exponents (4 and 5) are greater than radical index (3), so we can at least

extract one of those factors using power properties:

�24 ∙ 353 = (24 ∙ 35)13 = 2

4∙13 ∙ 3

5∙13 = 2

43 ∙ 3

53

Now we use the product power property to convert the improper fraction in a integer

plus a proper fraction

243 ∙ 3

53 = 2

33 ∙ 2

13 ∙ 3

33 ∙ 3

23 = 2 ∙ 3 ∙ 2

13 ∙ 3

23 = 6�2 ∙ 323 = 6√183

Example 2:

�𝑥𝑥8 ∙ 𝑦𝑦5 ∙ 𝑧𝑧34 = 𝑥𝑥2𝑦𝑦�𝑦𝑦𝑧𝑧34

84 = 2;

54 = 1 +

14

To introduce factors into a radical, do the inverse operation.


Unit 02 October

MATH VOCABULARY: Factor, To Factorize, To Extract, Equal, Improper Faction, Inverse.

4.4. ADDING AND SUBTRACTING RADICALS.

Two radicals can only be added (or subtracted) when they have the same index

and radicand, that is, when they are similar radicals.

3√5 + 2√5 − √5 = (3 + 2− 1)√5 = 4√5

2√12− 3√75 + √27 = 2�22 ∙ 3 − 3�3 ∙ 52 + �33 = 4√3− 15√3 + 3√3 = −8√3

MATH VOCABULARY: Similar Radicals.

4.5. MULTIPLYING AND DIVIDING RADICALS.

Due to the powers properties to multiply radicals with the same index, multiply

the radicands and the index remains the same.

√5 ∙ √10 = 512∙ ∙ 10

12 = (5 ∙ 10)

12 = √5 ∙ 10 = √50 = �2 ∙ 52 = 5√2

To multiply radicals with different index, reduce to a common index and then

multiply.

√𝑒𝑒 ∙ �𝑒𝑒34 = �𝑒𝑒48 ∙ �𝑒𝑒68 = �𝑒𝑒4 ∙ 𝑒𝑒68 = �𝑒𝑒108 = 𝑒𝑒�𝑒𝑒28 = 𝑒𝑒√𝑒𝑒4

To divide radicals we have to use the same rules, but dividing instead

multiplying.


Unit 02 October

4.6. POWERS AND ROOTS OF RADICALS.

To calculate the power or the root of a radical we express the radicals as

powers and we use the power of power property.

�√5�2

= �512�

2= 5

1∙22 = 5

�√234= �2

13

4

= �213�

14

= 2112 = √212

5. RATIONALIZING THE DENOMINATOR.

Sometimes in Algebra it is desirable to find an equivalent expression for a

radical expression that doesn’t have any radicals in the denominator. This process is

called rationalizing the denominator. We will use the multiplication identity property.

MATH VOCABULARY: To Rationalize, Algebra.

5.1. THE DENOMINATOR IS A SINGLE SQUARE ROOT.

When you have a single square root in the denominator you just multiply top

and bottom by it.

2√5

=2√5

∙√5√5

=2√5

5

5.2. THE DENOMINATOR IS A SINGLE NTH ROOT.

When you have a single nth root in the denominator, multiply by something so

that you will get a perfect nth power.

1√725 =

1√725 ∙

√735

√735 =√735

7


Unit 02 October

In general:

𝐚𝐚√𝐛𝐛𝐤𝐤𝐧𝐧 =

𝐚𝐚√𝐛𝐛𝐤𝐤𝐧𝐧 ∙

√𝐛𝐛𝐧𝐧−𝐤𝐤𝐧𝐧

√𝐛𝐛𝐧𝐧−𝐤𝐤𝐧𝐧 =𝐚𝐚√𝐛𝐛𝐧𝐧−𝐤𝐤𝐧𝐧

𝐛𝐛

5.3. THE DENOMINATOR IS EITHER A SUM OR A DIFFERENCE OF SQUARE ROOTS.

When you have a sum or a difference of square roots (Binomial) in the

denominator, multiply the top and the bottom by the conjugate of the denominator.

The conjugate of √𝐚𝐚 + √𝐛𝐛 is √𝐚𝐚 − √𝐛𝐛, and vice versa.

1√2− √3

=1

√2 − √3∙√2 + √3√2 + √3

=√2 + √3

2 − 3 = −√2 − √3

MATH VOCABULARY: Binomial, Conjugate.

6. SCIENTIFIC NOTATION.

Scientific Notation (also called Standard Form in Britain) is a special way of

writing numbers. It is a number of the form 𝐚𝐚 ∙ 𝟏𝟏𝟎𝟎𝐤𝐤, where:

• |𝐚𝐚| ∈ [𝟏𝟏,𝟏𝟏𝟎𝟎)

• 𝒌𝒌 ∈ ℤ, called Order of magnitude.

150,000,000 = 1.5 ∙ 108

0.0000081 = 8.1 ∙ 10−6

It is supposed you know already how to add, subtract, multiply and divide

numbers in scientific notation from previous courses

MATH VOCABULARY: Scientific Notation, Standard Form, Order of Magnitude.


Unit 02 October

7. LOGARITHMS.

Given two numbers, 𝒂𝒂 ∈ ℝ+,𝒂𝒂 ≠ 𝟏𝟏 and 𝐛𝐛 ∈ ℝ+, the logarithm in base 𝒂𝒂 of b,

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 𝐛𝐛, is the index we need for raising 𝒂𝒂 to get 𝒃𝒃.

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 𝐛𝐛 = 𝐯𝐯 𝐢𝐢𝐢𝐢 𝐚𝐚𝐯𝐯 = 𝐛𝐛

𝑒𝑒𝑡𝑡𝑙𝑙2 8 = 3 𝑏𝑏𝑒𝑒𝑏𝑏𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒 23 = 8

If the base is 10 (scientific notation), it is called Common Logarithm, and we

write it without the base.

log 100 = 2 𝑏𝑏𝑒𝑒𝑏𝑏𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒 102 = 100

If the base if the irrational number e, it is called Napierian logarithm. and we

write it 𝐞𝐞𝐧𝐧.

ln1𝑒𝑒3 = ln 𝑒𝑒−3 = −3

MATH VOCABULARY: Logarithm, To Raise, Common Logarithm, Napierian Logarithm.

8. LOGARITHMS PROPERTIES.

The logarithms have also some properties that will help us to solve exercises

and problems. All of them come from the power properties.

8.1. DIRECT PROPERTIES.

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 𝟏𝟏 = 𝟎𝟎, 𝐛𝐛𝐯𝐯𝐯𝐯𝐚𝐚𝐢𝐢𝐞𝐞𝐯𝐯𝐯𝐯 𝒂𝒂𝟎𝟎 = 𝟏𝟏

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 𝐚𝐚 = 𝟏𝟏, 𝐛𝐛𝐯𝐯𝐯𝐯𝐚𝐚𝐢𝐢𝐞𝐞𝐯𝐯𝐯𝐯 𝒂𝒂𝟏𝟏 = 𝒂𝒂


Unit 02 October

8.2. LOGARITHM OF A PRODUCT.

The logarithm of a product is the addition of the logarithms of each factor.

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚(𝐛𝐛 ∙ 𝐯𝐯) = 𝐞𝐞𝐭𝐭𝐠𝐠𝒂𝒂 𝒃𝒃 + 𝐞𝐞𝐭𝐭𝐠𝐠𝒂𝒂 𝒄𝒄

8.3. LOGARITHM OF A FRACTION.

The logarithm of a fraction is the subtraction of the logarithms of each factor.

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 �𝐛𝐛𝒄𝒄� = 𝐞𝐞𝐭𝐭𝐠𝐠𝒂𝒂 𝒃𝒃 − 𝐞𝐞𝐭𝐭𝐠𝐠𝒂𝒂 𝒄𝒄

8.4. LOGARITHM OF A POWER.

The logarithm of a power is the product of the index by the logarithm of the

power´s base.

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 𝐛𝐛𝒏𝒏 = 𝐧𝐧 ∙ 𝐞𝐞𝐭𝐭𝐠𝐠𝒂𝒂 𝒃𝒃

8.5. CHANGING BASES.

To change the base of a logarithm we use this formula:

𝐞𝐞𝐭𝐭𝐠𝐠𝐚𝐚 𝐛𝐛 =𝐞𝐞𝐭𝐭𝐠𝐠𝒄𝒄 𝒃𝒃𝐞𝐞𝐭𝐭𝐠𝐠𝒄𝒄 𝒂𝒂

𝑒𝑒𝑡𝑡𝑙𝑙4 12 =log 12log 4


Unit 02 October

8.6. LOGARITHMS EQUATIONS.

To solve logarithms equations we have to apply the logarithms and power

properties.

𝑒𝑒𝑡𝑡𝑙𝑙9 𝑥𝑥 =14 ⇒ 9

14 = 𝑥𝑥 ⇒ (32)

14 = 3

24 = 3

12 = √3 = 𝑥𝑥


Unit 02 October

1. Number each of the boxes below, 1-16. On a separate sheet of paper, show

work for each box as you solve the logarithm equation. Each box has at least

one equation. Show all your work.

2. On the puzzle below, write your answer for x next to each equation. Then cut

out each box individually.

3. To “solve” the puzzle, the touching edges should be equivalent.

For example, 2log 4x = should touch the edge where 16x = .

4. Glue down your pieces on a separate sheet of paper. The final product you

turn in should be the re-arranged puzzle pieces (should still look like a square)

and the page of work you did for all 16 individual squares.


4 ESO Academics - UNIT 02 - POWERS, ROOTS AND LOGARITHMS

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Transcript of 4 ESO Academics - UNIT 02 - POWERS, ROOTS AND LOGARITHMS