Date: Mon, 25 Sep 2023 20:18:57 +0700 Subject: Unlocked website, added rationalising roots --- lessons-locked/matematika/pangkat-akar/akar.html | 124 --------------- .../matematika/pangkat-akar/latihan.html | 167 --------------------- .../matematika/pangkat-akar/pangkat.html | 139 ----------------- .../matematika/pangkat-akar/scientific.html | 69 --------- 4 files changed, 499 deletions(-) delete mode 100644 lessons-locked/matematika/pangkat-akar/akar.html delete mode 100644 lessons-locked/matematika/pangkat-akar/latihan.html delete mode 100644 lessons-locked/matematika/pangkat-akar/pangkat.html delete mode 100644 lessons-locked/matematika/pangkat-akar/scientific.html (limited to 'lessons-locked/matematika/pangkat-akar') diff --git a/lessons-locked/matematika/pangkat-akar/akar.html b/lessons-locked/matematika/pangkat-akar/akar.html deleted file mode 100644 index e7505de..0000000 --- a/lessons-locked/matematika/pangkat-akar/akar.html +++ /dev/null @@ -1,124 +0,0 @@ - - - - - - - - - - Al Azhar 9th Grade Lesson Notes - - - - - - - - -

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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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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