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Chapter 1 - Bentuk Pangkat dan Akar

Lesson 1: Exponent

Lesson 2: Root

Exercise 1

Lesson 3: Exponents Extended - Scientific Form of Decimals

Chapter 2 - Quadratic Equation

Main Lesson: Quadratic Equation - The Introduction

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Definition

- Root is the inverse of exponent. For example;
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4² = 16, so 16 = 4
5² = 25, so 25 = 5
10² = 100, so 100 = 10

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Simplifying Square Roots

- You can simplify a root by factoring the root number another number that is rational. Or in other words, divide the root number by the perfect squares. For example; -

- Simplify 60
- 60 = 4 × 15 = 215
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- Simplify 48
- 48 = 16 × 3 = 43
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- Simplify 125
- 125 = 25 × 5 = 55
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Root Operations

Addition

- Roots with coefficient can be added with another coefficient that have the same roots.
- ac + bc = (a + b)c -

32 + 42 = (3 + 4)2 = 72
5 + 35 = (1 + 3)5 = 45

Subtraction

- Roots with coefficient can be subtracted with another coefficient that have the same roots.
- ac - bc = (a - b)c -

73 - 43 = (7 - 4)3 = 32
106 - 56 = (10 - 5)6 = 56

Multiplication

- Roots can be multiplied with any other roots. If the root is the same as the other multiplied root, the root can be deleted (if you want to be faster).
- ac × bd = (a × b)c × d -

2 × 5 = 10
32 × 43 = (3 × 4)2 × 3 = 126
25 × 35 = (2 × 3)(5) = 6(5) = 30

Division

- Roots can be divided with any other roots.
- ac ÷ bd = (a ÷ b)c ÷ d -

- 86 ÷ 43 = (8 ÷ 4)6 ÷ 3 = 22 -

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Pertanyaan

Pangkat

8p²q × 2pq³
64^¹⁄₂ + 81^¹⁄₃
25 + 320 - 45

Merasionalkan Akar

⁴⁄₅
³⁵⁄₂₃
⁵⁄_{3 - 2}
^{5 - 7}⁄_{5 - 2}

Operasi Pangkat

(8p²3)³ ÷ 2pq
^{(a^¹⁄₃b^¹⁄₃)⁶}⁄_{(a^²⁄₃b^⁴⁄₃)⁹}

Persamaan Pangkat

2^{x + 8} = 64
16^{2x + 3} = 32

Jawaban

16p³q⁴
- 64^¹⁄₂ + 81^¹⁄₃
- = 64 + ³81
- = 8 + ³81 -
- 25 + 320 - 45
- = 65 + 320
- = 65 + 34 × 5
- = 65 + 32² × 5
- = 65 + 3 × 25
- = 65 + 65
- = 125
-
- ⁴⁄₅
- = ⁴⁄₅ × ⁵⁄₅ = ⁴⁵⁄₅ -
- ³⁵⁄₂₃
- = ³⁵⁄₂₃ × ³⁄₃
- = ³¹⁵⁄_{2 × 3}
- = ¹⁵⁄₂ -
- ⁵⁄_{3 - 2}
- = ⁵⁄_{3 - 2} × ^{3 + 2}⁄_{3 + 2}
- = ^{5(3 + 2)}⁄_{(3)² - (2)²}
- = ^{5(3 + 2)}⁄_{3 - 2} = ^{5(3 + 2)}⁄₁
- = 5(3 + 2) -
- ^{5 - 7}⁄_{5 - 2}
- = ^{5 - 7}⁄_{5 - 2} × ^{5 + 2}⁄_{5 + 2}
- = ^{(5 - 7) × (5 + 2)}⁄_{(5)² - (2)²}
- = ^{5 + 10 - 35 - 14}⁄_{5 - 2} = ^{5 + 10 - 35 - 14}⁄₃ -
- (8p²3)³ ÷ 2pq
- = (24p²)³ ÷ 2pq
- = 24³ p⁶ ÷ 2pq
- = 6912p⁵ ÷ q -
- = ^{(a^¹⁄₃b^¹⁄₃)⁶}⁄_{(a^²⁄₃b^⁴⁄₃)⁹}
- = - ^{(a^¹⁄₃)⁶ (b^¹⁄₃)⁶} - ⁄ - _{(a^²⁄₃)⁹ (b^⁴⁄₃)⁹} -
- = - ^{a² b²} - ⁄ - _{a⁶ b¹²} -
- = a^-4 b^-10 -
- 2^{x + 8} = 64
- = 2^{x + 8} = 2⁶
- = x + 8 = 6
- = x = 6 - 8
- = x = -2 -
- 16^{2x + 3} = 32
- = (2⁴)^{2x + 3} = 2⁶
- = 2^{8x + 12} = 2⁶
- = 8x + 12 = 6
- = 8x = 6 - 12
- = 8x = 6
- = x = ^-6⁄₈
- = x = -³⁄₄ -

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Exponents

- xⁿ = x₁ × x₂ × ... × x_n -

Examples

2⁵ = 2 × 2 × 2 × 2 × 2 = 32
5⁷ = 5 × 5 × 5 × 5 × 5 × 5 × 5 = 78.125

Exponent Operations

Multiplication

- a^x × b^y = (a × b)^{x + y} -

10⁴ × 2² = (10 × 2)^{4 + 2} = 20⁶

Division

- a^x ÷ b^y = (a ÷ b)^{x - y} -

10³ ÷ 2⁴ = (10 ÷ 2)^{4 - 2} = 5²

Exponent

- (a^x)^y = a^{x × y} -

(2²)³ = 2^{2 × 3} = 2⁶ = 64

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Zero Exponent

a⁰ = 1

Examples

5⁰ = 1
7⁰ = 1
100⁰ = 1
28,67⁰ = 1

One Exponent

a¹ = a

Examples

5¹ = 5
7¹ = 7
100¹ = 100
28,67¹ = 28,67

Fractional Exponent

x ^{^y⁄_z} = - ^zx^y

Examples

5 ^¹⁄₂ = - ²5
7 ^²⁄₃ = - ³7²

Small Note: 2x or x menans square root. The 2 before root should be removed. The 2 is only present for easier purposes.
3x means cubic root.

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Exponential Equation

2^{x + 5} = 64
- = 2^{x + 5} = 2⁶ _{↖ ubah ke bentuk 2 berpangkat (2^x)}
- = x + 5 = 6
- = x = 1 -
16^{x + 2} = 32
- = (2⁴)^{x + 2} = 2⁵
- = 2^{4x + 8} = 2⁵
- = 4x + 8 = 5
- = 4x = 5 - 8
- = 4x = -3
- = x = ^-3⁄₄ = -³⁄₄ -

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- Firstly, lets use this equation.
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Step-by-step

Find the factors of c, which in this equation, c = 30.
The factors of 30 are: -
- 1, 30
- 2, 15
- 3, 10
- 5, 6
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Lets define that the first and second factors are y_{1, 2}. To find the suitable factors for the equation, you need to check if; -
- , and
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Check for each of the factors. We end up with 5 and 6.
So, the answer is 5 & 6

← Go back to Main Lesson

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Introduction

- In algebra, a quadratic equation is any equation that can be rearranged in standard form as where x - represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0.^[1] -

Formula / Format

- ax² + bx + c = 0 -

- a, b, c = known numbers, where a ≠ 0
- x = the unknown, or the root -

Exercises

Finding the roots with factorisation.

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- To find the answer, we need to find the factors of c that is suitable for b and c. What and how do I find the suitable factors?
- The suitable factor for the equation are 2 and -3.
- Then, insert each numbers to this formula: , where y is the factor.

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- So, the answer is -
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- Factors: 2, 4
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Relation between Quadratic Equation and Algebraic Expressions

This is an algebraic expression → . You can turn that into a quadratic equation.

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Quadratic Equation's Root Operations

Finding Quadratic Equation with Roots

You can use the formula below to find a Quadratic Equation with 2 roots.

- , where x_{1, 2} are the roots.
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- Example:

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Quadratic Root's Mathematical Operations

Formulas

Examples

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Diketahui persamaan kuadrat , memiliki akar-akar persamaan kuadrat x₁ dan x₂. Jika x₁ < x₂, maka tentukan: -
1. x₁ + x₂
2. x₁ × x₂
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References

Quadratic Equation - The English Wikipedia, https://en.wikipedia.org/wiki/Quadratic_equation

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Al Azhar 9th Grade Lesson Notes

Table of Contents

Matematika

Hidayatullah, S.Pd.

Chapter 1 - Bentuk Pangkat dan Akar

Chapter 2 - Quadratic Equation

Al Azhar 9th Grade Lesson Notes

Table of Contents

Matematika

Al Azhar 9th Grade Lesson Notes

Table of Contents

Root

Chapter 1

Definition

Simplifying Square Roots

Root Operations

Addition

Subtraction

Multiplication

Division

Al Azhar 9th Grade Lesson Notes

Table of Contents

Exercise

Chapter 1

Pertanyaan

Pangkat

Merasionalkan Akar

Operasi Pangkat

Persamaan Pangkat

Jawaban

Al Azhar 9th Grade Lesson Notes

Table of Contents

Exponents

Chapter 1

Exponents

Examples

Exponent Operations

Multiplication

Division

Exponent

Zero Exponent

Examples

One Exponent

Examples

Fractional Exponent

Examples

Exponential Equation

Al Azhar 9th Grade Lesson Notes

Table of Contents

Standard Form (decimals)

Chapter 1

Al Azhar 9th Grade Lesson Notes

Table of Contents

Quadratic Equation: Finding the suitable factor

Chapter 2 sub-page

Step-by-step

Al Azhar 9th Grade Lesson Notes

Table of Contents

Quadratic Equation: The Introduction

Chapter 2

Introduction

Formula / Format

Exercises

Finding the roots with factorisation.

Relation between Quadratic Equation and Algebraic Expressions

Quadratic Equation's Root Operations

Finding Quadratic Equation with Roots

Quadratic Root's Mathematical Operations

Formulas

Examples

References