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Index Notation
A way of representing numbers (constants) and variables (e.g.x x and y) that have been
multiplied by themselves a number of times.
Examples:
ξ₯΅
οΈ
ξ‘ ξ΄½
οΉ
ξ‘ ξ₯΄ξ΅
ο»
ξ‘
σ°
ξ₯Άξ΅
οΆ
ξ΅
οΈ
σ°
ο»
ξ‘ ξ΅
οΏοΉ
οΆ
Term:
A single number (constant) or variable
E.g.,
In the expression 4x β 7 both 4x and β7 are terms.
Coefficients:
The number which the variable is being multiplied by
E.g.,
In 2x
3
the coefficient is 2.
Integer:
A whole number
E.g.,
1, 7 and 1003.
Index (also called exponent or powers):
The index number is the number of times you multiply a number/variable by itself. It is normally
E.g.,
The index number in 5
4
is 4.
Note: the plural of index is indices
Note: you will see index number as a superscript
Base Number:
The number/unknown that is being multiplied by itself an amount of times
E.g.,
The base number in 5
4
is 5 and in 2x
3
the base number is x.
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Square Numbers:
A number/variable that is βsquaredβ is multiplied by itself
E.g.
4 Γ 4 can be written as 4
2
and is spoken as β4 squaredβ or β4 to the power of 2β.
A square number is found when we multiply an integer (whole number) by itself.
E.g.
1Γ1=1Therefore 1 is a square number2Γ2=4Therefore 4 is a square
number3Γ3=9Therefore 9 is a square number10Γ10=100Therefore 100 is a square
number1Γ1=1Therefore 1 is a square number2Γ2=4Therefore 4 is a square
number3Γ3=9Therefore 9 is a square number10Γ10=100Therefore 100 is a square
number
Cube Number:
A number/variable that is βcubedβ is multiplied by itself three times.
E.g.
4 Γ 4 Γ 4 can be written as 4
3
and is spoken as β4 cubedβ.
A cube number is found when we multiply an integer (whole number) by itself three times.
E.g.
1Γ1Γ1=1Therefore 1 is a cube number2Γ2Γ2=8Therefore 8 is a cube
number3Γ3Γ3=27 Therefore 27 is a cube number10Γ10Γ10=1000Therefore 1000 is
a cube number
How to simplify expressions involving index notation
1. Identify whether the base numbers for each term are the same
In higher tier questions you may need to manipulate the base numbers first
2. Identify the operation/s being undertaken between the terms
3. Follow the rules of index notation to simplifying the expression
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How to use the laws of indices
We can use the laws of indices to simplify expressions.
Law 1: multiplying indices
When multiplying indices with the same base, add the powers.
2 Multiply any coefficients.
Example:
Simplify each of the following, leaving your answer in index notation where appropriate.
1. a. ξ₯΄
ο·
ξ΅ξξ₯΄
ο»
This can be written as
ξ₯΄ξξ΅ξξ₯΄ξξ΅ξ₯΄ξξ΅ξξ₯΄ξξ΅ξξ₯΄ξξ΅ξξ₯΄ξξ΅ξξ₯΄ξξ΅ξξ₯΄ξξ΅ξξ₯΄ξξ΅ξξ₯΄ξ
You will notice that we multiplied 2 ten times.
Therefore, ξ₯΄
ο·
ξ΅ξξ₯΄
ο»
ο ξ₯΄
ο΅ο΄
Check:
ξ₯΄
ο·
ο ξ₯Ίξξ΄½ξ΅ξ΅ξξ₯΄
ο»
ο ξ₯³ξ₯΄ξ₯Ί
Multiply them and you will get ξ₯Ίξξ΅ξξ₯³ξ₯΄ξ₯Ί ο ξ₯³ξ₯²ξ₯΄ξ₯Ά
ξ₯΄
ο΅ο΄
ο ξ₯³ξ₯²ξ₯΄ξ₯Ά
If both terms have the same base, we can simply write it as ξ₯΄
ο·
ξ΅ξξ₯΄
ο»
ο ξ₯΄
ο·οΎο»
ο ξ₯΄
ο΅ο΄
b. σ°οξ₯Άσ°
οΊ
ξ΅ξσ°οξ₯Άσ°
οΉ
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This can be written as
σ°
οξ₯Ά
σ°
ξξ΅ξ
σ°
οξ₯Ά
σ°
ξξ΅ξ
σ°
οξ₯Ά
σ°
ξξ΅ξ
σ°
οξ₯Ά
σ°
ξξ΅ξ
σ°
οξ₯Ά
σ°
ξξ΅ξ
σ°
οξ₯Ά
σ°
ξξ΅ξ
σ°
οξ₯Ά
σ°
ξ΅ξ
σ°
οξ₯Ά
σ°
ξ΅ξ
σ°
οξ₯Ά
σ°
ξ΅ξ
σ°
οξ₯Ά
σ°
ξ΅ξ
σ°
οξ₯Ά
σ°
You will notice that we multiplied (-4) eleven times.
Therefore, σ°οξ₯Άσ°
οΊ
ξ΅ξ
σ°
οξ₯Ά
σ°
οΉ
ο ξ₯Ά
ο΅ο΅
Check:
σ°οξ₯Άσ°
οΊ
ο ξ₯Άξ₯²ξ₯»ξ₯Έξξ΄½ξ΅ξ΅ξσ°οξ₯Άσ°
οΉ
ο οξ₯³ξ₯²ξ₯΄ξ₯Ά
Multiply them and you will get
ξ₯Άξ₯²ξ₯»ξ₯Έξξ΅ ο ξ₯³ξ₯²ξ₯΄ξ₯Ά ο οξ₯Άξ‘ξ₯³ξ₯»ξ₯Άξ‘ξ₯΅ξ₯²ξ₯Ά
σ°οξ₯Άσ°
ο΅ο΅
ο οξ₯Άξ‘ξ₯³ξ₯»ξ₯Άξ‘ξ₯΅ξ₯²ξ₯Ά
If both terms have the same base, we can simply write it as
σ°οξ₯Άσ°
οΊ
ξ΅ξσ°οξ₯Άσ°
οΉ
ο σ°οξ₯Άσ°
οΊοΎοΉ
ο σ°οξ₯Άσ°
ο΅ο΅
c. ξ΅
οΌ
ξ΅ξξ΅
ο·
This can be written as
ξΊξξξξΊξξξξΊξξξξΊξξξξΊξξξξΊξξξξΊξξξξΊξξξξΊξξξξΊξξξξΊξξ
You will notice that we multiplied x eleven times.
Therefore, ξΊ
ο‘
ξξξΊ
ο
ο ξ΅
ο΅ο΅
If both terms have the same base, we can simply write it as
ξ΅
οΌ
ξ΅ξξ΅
ο·
ο ξ΅
οΌοΎο·
ο ξ΅
ο΅ο΅
d.
σ°
ξ₯΅ξ΅
οΆ
σ°
ξ΅ξσ°ξ₯Ίξ΅
ο»
σ°
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This can be written as
σ°
ξ₯΅ξ΅
οΆ
σ°
ξξξξξξξξξξξξξξξξξξξ₯΅ξξξξ΅ξξξξ΅
σ°
ξ₯Ίξ΅
ο»
σ°
ξξξξξξξξξξξξξξξξξξ₯Ίξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξ
We can write is as
ξ₯΅ξξξξ₯Ίξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅
You will multiply the coefficients, 3 and 8 and add the indices 2 and 7
Therefore ξ₯΅ξξξξ₯Ίξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ξξξξ΅ ο ξ₯΄ξ₯Άξξξξ΅
ο½
or ξ₯΄ξ₯Άξ΅
ο½
If both terms have the different base, we can simply write it as
σ°ξ₯΅ξ΅
οΆ
σ°ξξ΅ξσ°ξ₯Ίξ΅
ο»
σ° ο
σ°
ξ₯΅ξξ΅ξξ₯Ί
σ°
ξ΅ξξ΅
οΆοΎο»
ο ξ₯΄ξ₯Ά
ο½
Law 2: dividing indices
When dividing indices with the same base, subtract the powers.
1 Subtract the indices.
2 Divide any coefficients of the base number or letter.
Example:
Simplify each of the following, leaving your answer in index notation where appropriate
2. a. ξ₯·
οΌ
ο ξ₯·
οΉ
Identify the indices and subtract: 8 and 5, 8 β 5 = 3
We can write it as
οΉξο«ξοΉξο«ξοΉξο«ξοΉξξο«ξοΉξο«ξοΉξο«ξοΉξο«ξοΉ
οΉξο«ξοΉξο«ξοΉξο«ξοΉξο«ξοΉ
Dividing them will give us
οΉξο«ξοΉξο«ξοΉξο«ξοΉξξο«ξοΉξο«ξοΉξο«ξοΉξο«ξοΉ
οΉξο«ξοΉξο«ξοΉξο«ξοΉξο«ξοΉ
ο ξξξ₯·ξξ΅ξξ₯·ξξ΅ξξ₯·ξξ΅ξ΅ξξ₯·
ο·
Therefore, ξ₯·
οΌ
ο ξ₯·
οΉ
ο ξ₯·
ο·
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If both terms have the same base, we can simply write it as
ξ₯·
οΌ
ο ξ₯·
οΉ
ο ξ₯·
οΌοΏοΉ
ο ξ₯·
ο·
b. σ°οξ₯Ήσ°
ο΅ο΅
ο σ°οξ₯Ήσ°
οΈ
Identify the indices and subtract: 11 and 4, 11 β 4 = 7
We can write it as
σ°
οΏο»
σ°
ξο«ξ
σ°
οΏο»
σ°
ξο«ξ
σ°
οΏο»
σ°
ξο«ξ
σ°
οΏο»
σ°
ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°
σ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°
Dividing them will give us
σ°
οΏο»
σ°
ξο«ξ
σ°
οΏο»
σ°
ξο«ξ
σ°
οΏο»
σ°
ξο«ξ
σ°
οΏο»
σ°
ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°
σ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°ξο«ξσ°οΏο»σ°
ξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξο ξξξ
σ°
οξ₯Ή
σ°
ξξ΅
σ°
οξ₯Ή
σ°
ξ΅
σ°
οξ₯Ή
σ°
ξ΅ξ
σ°
οξ₯Ή
σ°
ξ΅ξ
σ°
οξ₯Ή
σ°
ξξ΅ξσ°οξ₯Ήσ°ξξ΅ξσ°οξ₯Ήσ°ξξ΅ξ΅ξσ°οξ₯Ήσ°
ο»
Therefore, σ°οξ₯Ήσ°
ο΅ο΅
ο σ°οξ₯Ήσ°
οΈ
ο σ°οξ₯Ήσ°
ο»
If both terms have the same base, we can simply write it as
σ°οξ₯Ήσ°
ο΅ο΅
ο σ°οξ₯Ήσ°
οΈ
ο σ°οξ₯Ήσ°
ο΅ο΅οΏοΈ
ο σ°οξ₯Ήσ°
ο»
c. ξ₯Έξ΅
ο»
ο ξ΅
ο·
Identify the indices and subtract: 7 and 3, 7 β 3 = 4
We can write it as
οΊξξο«ξξ΅ξο«ξξ΅ξο«ξξ΅ξξο«ξξ΅ξο«ξξ΅ξο«ξξ΅ξο«ξξ΅ξ
ξ΅ξο«ξξ΅ξο«ξξ΅ξ
Dividing them will give us ξ
ξ₯Έξξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξ
ξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξ
If there is no coefficient, then that means that the coefficient is 1.
ξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξο ξξξξ₯Έξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξξ΅ξ΅ξξ₯Έξ΅
οΈ
Therefore, ξ₯Έξ΅
ο»
ο ξ΅
ο·
ο ξ₯Έξ΅
οΈ
If both terms have the different base, we can write it as
ξ₯Έξ΅
ο»
ο ξ΅
ο·
ο
σ°
ξ₯Έ ο ξ₯³
σ°
ξ΅ξσ°ξ΅σ°
ο»οΏο·
ο ξ₯Έξξξξ΅
οΈξ
ο ξ₯Έξ΅
οΈ
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d.
σ°
οξ₯³ξ₯·ξ΅
ο½
σ°
ο ξ₯·ξ΅
οΈ
Identify the indices and subtract: 9 and 4, 9 β 4 = 5
We can write it as
οΏο΅οΉξξο«ξο¬ξο«ξο¬ξο«ξο¬ξξο«ξο¬ξο«ξο¬ξο«ξο¬ξο«ξο¬ξο«ξο¬ξο«ξο¬ξ
οΉξο«ξο¬ξο«ξο¬ξο«ξο¬ξο«ξο¬ξ
Dividing them will give us ξ
οξ₯³ξ₯·ξξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξ
ξ₯·ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξ
ξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξξο ξξ οξ₯΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξ΅ξξξ΅ξ΅ξοξ₯΅ξ΅
οΉ
Therefore, σ°οξ₯³ξ₯·ξ΅
ο½
σ° ο ξ₯·ξ΅
οΈ
ο οξ₯΅ξ΅
οΉ
If both terms have the different base, we can write it as
σ°οξ₯³ξ₯·ξ΅
ο½
σ° ο ξ₯·ξ΅
οΈ
ο
σ°
οξ₯³ξ₯· ο ξ₯·
σ°
ξ΅ξσ°ξ΅σ°
ο½οΏοΈ
ο οξ₯΅ξξξξ΅
οΉξ
ο οξ₯΅ξ΅
οΉ

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Index Notation A way of representing numbers (constants) and variables (e.g.x x and y) that have been multiplied by themselves a number of times. Examples: 34 , π5 , 2π₯ 7 , (4π¦ 2 π₯ 4 )7 , π§ β5 2 Term: A single number (constant) or variable E.g., In the expression 4x β 7 both 4x and β7 are terms. Coefficients: The number which the variable is being multiplied by E.g., In 2x3 the coefficient is 2. Integer: A whole number E.g., 1, 7 and 1003. Index (also called exponent or powers): The index number is the number of times you multiply a number/variable by itself. It is normally shown about the base number E.g., The index number in 54 is 4. Note: the plural of index is indices Note: you will see index number as a superscript Base Number: The number/unknown that is being multiplied by itself an amount of times E.g., The base number in 54 is 5 and in 2x3 the base number is x. Square Numbers: A number/variable that is βsquaredβ is multiplied by itself E.g. 4 Γ 4 can be written as 42 and is spoken as β4 squaredβ or β4 to the power of 2β. A square number is found when we multiply an integer (whole number) by itself. E.g. 1Γ1=1Therefore 1 is a square number2Γ2=4Therefore 4 is a square number3Γ3=9Therefore 9 is a square number10Γ10=100Therefore 100 is a square number1Γ1=1Therefore 1 is a square number2Γ2=4Therefore 4 is a square number3Γ3=9Therefore 9 is a square number10Γ10=100Therefore 100 is a square number Cube Number: A number/variable that is βcubedβ is multiplied by itself three times. E.g. 4 Γ 4 Γ 4 can be written as 43 and is spoken as β4 cubedβ. A cube number is found when we multiply an integer (whole number) by itself three times. E.g. 1Γ1Γ1=1Therefore 1 is a cube number2Γ2Γ2=8Therefore 8 is a cube number3Γ3Γ3=27 Therefore 27 is a cube number10Γ10Γ10=1000Therefore 1000 is a cube number How to simplify expressions involving index notation 1. Identify whether the base numbers for each term are the same In higher tier questions you may need to manipulate the base numbers first 2. Identify the operation/s being undertaken between the terms 3. Follow the rules of index notation to simplifying the expression How to use the laws of indices We can use the laws of indices to simplify expressions. Law 1: multiplying indices When multiplying indices with the same base, add the powers. 1 Add the powers. 2 Multiply any coefficients. Example: Simplify each of the following, leaving your answer in index notation where appropriate. 1. a. 23 π₯ 27 This can be written as 2 π₯ 2 π₯2 π₯ 2 π₯ 2 π₯ 2 π₯ 2 π₯ 2 π₯ 2 π₯ 2 You will notice that we multiplied 2 ten times. Therefore, 23 π₯ 27 = 210 Check: 23 = 8 πππ 27 = 128 Multiply them and you will get 8 π₯ 128 = 1024 210 = 1024 If both terms have the same base, we can simply write it as 23 π₯ 27 = 23+7 = 210 b. (β4)6 π₯ (β4)5 This can be written as (β4) π₯ (β4) π₯ (β4) π₯ (β4) π₯ (β4) π₯ (β4) π₯ (β4)π₯ (β4)π₯ (β4)π₯ (β4)π₯ (β4) You will notice that we multiplied (-4) eleven times. Therefore, (β4)6 π₯ (β4)5 = 411 Check: (β4)6 = 4096 πππ (β4)5 = β1024 Multiply them and you will get 4096 π₯ β 1024 = β4,194,304 (β4)11 = β4,194,304 If both terms have the same base, we can simply write it as (β4)6 π₯ (β4)5 = (β4)6+5 = (β4)11 c. π₯ 8 π₯ π₯ 3 This can be written as πxπxπxπxπxπxπxπxπxπxπ You will notice that we multiplied x eleven times. Therefore, ππ x ππ = π₯11 If both terms have the same base, we can simply write it as π₯ 8 π₯ π₯ 3 = π₯ 8+3 = π₯ 11 d. (3π¦ 2 )π₯ (8π¦ 7 ) This can be written as (3π¦ 2 ) is equivalent to 3 x π¦ x π¦ (8π¦ 7 )is equivalent to 8 x π¦ x π¦ x π¦ x π¦ x π¦ x π¦ x π¦ We can write is as 3x8xπ¦xπ¦xπ¦xπ¦xπ¦xπ¦xπ¦xπ¦xπ¦ You will multiply the coefficients, 3 and 8 and add the indices 2 and 7 Therefore 3 x 8 x π¦ x π¦ x π¦ x π¦ x π¦ x π¦ x π¦ x π¦ x π¦ = 24 x π¦ 9 or 24π¦ 9 If both terms have the different base, we can simply write it as (3π¦ 2 ) π₯ (8π¦ 7 ) = (3 π₯ 8)π₯ π¦ 2+7 = 249 Law 2: dividing indices When dividing indices with the same base, subtract the powers. 1 Subtract the indices. 2 Divide any coefficients of the base number or letter. Example: Simplify each of the following, leaving your answer in index notation where appropriate 2. a. 58 Γ· 55 Identify the indices and subtract: 8 and 5, 8 β 5 = 3 We can write it as 5π₯5π₯5π₯5 π₯5π₯5π₯5π₯5 5π₯5π₯5π₯5π₯5 Dividing them will give us Therefore, 58 Γ· 55 = 53 5π₯5π₯5π₯5 π₯5π₯5π₯5π₯5 5π₯5π₯5π₯5π₯5 = 5 π₯ 5 π₯ 5 ππ 53 If both terms have the same base, we can simply write it as 58 Γ· 55 = 58β5 = 53 b. (β7)11 Γ· (β7)4 Identify the indices and subtract: 11 and 4, 11 β 4 = 7 We can write it as (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) (β7) π₯ (β7) π₯ (β7) π₯ (β7) Dividing them will give us (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) π₯ (β7) (β7) π₯ (β7) π₯ (β7) π₯ (β7) = (β7) π₯(β7)π₯(β7)π₯ (β7)π₯ (β7) π₯ (β7) π₯ (β7) ππ (β7)7 Therefore, (β7)11 Γ· (β7)4 = (β7)7 If both terms have the same base, we can simply write it as (β7)11 Γ· (β7)4 = (β7)11β4 = (β7)7 c. 6π₯ 7 Γ· π₯ 3 Identify the indices and subtract: 7 and 3, 7 β 3 = 4 6 π₯π₯π₯π₯π₯π₯ π₯π₯π₯π₯π₯π₯π₯π₯ We can write it as π₯π₯ π₯π₯π₯ Dividing them will give us 6 π₯ π₯π₯π₯π₯π₯ π₯π₯π₯π₯π₯π₯π₯π₯ π₯π₯ π₯π₯π₯ If there is no coefficient, then that means that the coefficient is 1. = 6 π₯ π₯ π₯ π₯ π₯ π₯ π₯ π₯ ππ 6π₯ 4 Therefore, 6π₯ 7 Γ· π₯ 3 = 6π₯ 4 If both terms have the different base, we can write it as 6π₯ 7 Γ· π₯ 3 = (6 Γ· 1)π₯ (π₯)7β3 = 6 x π₯ 4 = 6π₯ 4 d. (β15π¦ 9 ) Γ· 5π¦ 4 Identify the indices and subtract: 9 and 4, 9 β 4 = 5 We can write it as β15 π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ Dividing them will give us 5π₯π¦π₯π¦π₯π¦π₯π¦ β15 π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ 5π₯π¦π₯π¦π₯π¦π₯π¦ = β3 π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ π₯ π¦ ππ β3π¦ 5 Therefore, (β15π¦ 9 ) Γ· 5π¦ 4 = β3π¦ 5 If both terms have the different base, we can write it as (β15π¦ 9 ) Γ· 5π¦ 4 = (β15 Γ· 5)π₯ (π¦)9β4 = β3 x π¦ 5 = β3π¦ 5 Name: Description: ...
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