diff options
Diffstat (limited to 'lessons/matematika')
| -rw-r--r-- | lessons/matematika/1-exponents/1-id.md | 77 | ||||
| -rw-r--r-- | lessons/matematika/1-exponents/1.md | 41 | ||||
| -rw-r--r-- | lessons/matematika/1-exponents/2-id.md | 177 | ||||
| -rw-r--r-- | lessons/matematika/1-exponents/2.md | 176 | ||||
| -rw-r--r-- | lessons/matematika/1-exponents/3-id.json | 0 | ||||
| -rw-r--r-- | lessons/matematika/1-exponents/3.json | 24 |
6 files changed, 476 insertions, 19 deletions
diff --git a/lessons/matematika/1-exponents/1-id.md b/lessons/matematika/1-exponents/1-id.md index d791bd0..05a5d69 100644 --- a/lessons/matematika/1-exponents/1-id.md +++ b/lessons/matematika/1-exponents/1-id.md @@ -1,19 +1,74 @@ # Pangkat \\( x^n = x_1 \times x_2 \times\ \cdot\cdot\cdot\ \times x_n \\) -!---( -#--( -## 💡 Loh, kenapa? -)--# - -Ini **harus** menjadi bagian vital dari website ini. -Menurutku, masalah utama di kurikulum Indonesia ialah kurangnya penjelasan MENGAPA, hanya APA. - -)---! - ## Contoh * \\(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \\) * \\(5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78125 \\) -## Operasi Berpangkat +# Operasi Berpangkat +## Perkalian +Jika sebuah angka berpangkat dikalikan dengan angka berpangkat lainnya, pangkat +kedua angka tersebut akan ditambah. + +\\( a^x \times b^y = (a \times b)^{x + y} \\) +* \\( 10^4 \times 2^2 = (10 \times 2)^{4 + 2} = 20^6 \\) + +## Pembagian +Jika sebuah angka berpangkat dibagi dengan angka berpangkat lainnya, pangkat +kedua angka tersebut akan dikurangi. + +\\( a^x \div b^y = (a \div b)^{x - y} \\) +* \\( 10^3 \div 2^4 = (10 \div 2)^{(4 - 2)} = 5^2 = 25 \\) + +## Pangkat +Jika sebuah angka berpangkat dipangkat, pangkatnya akan dikalikan. + +\\( (a^x)^y = a^{x \times y} \\) +* \\( (2^2)^3 = 2^{2 \times 3} = 2^6 = 64 \\) + +# Pangkat Istimewa +Beberapa pangkat merasa mereka lebih baik dari yang lain, dan Matematika setuju +dengan mereka. + +## Pangkat Nol +If a number is raised to the power of **zero**, the result is **one**. +Jika sebuah angka dipangkatkan ke **nol**, hasilnya akan selalu **satu**. + +\\( a^0 = 1 \\) +* \\( 5^0 = 1 \\) +* \\( 69.420^0 = 1 \\) + +## Pangkat Negatif +Jika sebuah angka dipangkatkan dengan angka negatif, juga dikatakan dengan +bilangan kebalikan, akan sama dengan... + +\\( a^{-b} = \dfrac{1}{a^b} \\) + +## Pangkat Pecahan +Akar **Penyebut (y)** dari **bilangan(a)** yang dipangkatkan dengan **Pembilang (x)**. + +\\( a^{\frac{x}{y}} = \sqrt[y]{a^x} \\) +* \\( 5^{\frac{1}{2}} = \sqrt[2]{5} \\) +* \\( 7^{\frac{2}{3}} = \sqrt[3]{7^2} \\) +## Persamaan Berpangkat +1. $$ +\begin{aligned} +2^{x + 5} &= 64 \\\\ +2^{x + 5} &= 2^6 &\leftarrow\textnormal{ kamu bisa ganti 64 ke bentuk 2 berpangkat (}2^x\textnormal{)} \\\\ +x + 5 &= 6 &\textnormal{eliminasi angka 2nya} \\\\ +x &= 6 - 5 \\\\ +&= 1 +\end{aligned} +$$ +1. $$ +\begin{aligned} +16^{x + 2} &= 2^5 \\\\ +(2^4)^{x + 2} &= 2^5 \\\\ +2^{4x + 8} &= 2^5 \\\\ +4x + 8 &= 5 \\\\ +4x &= 5 - 8 \\\\ +4x &= -3 \\\\ +x &= -\dfrac{3}{4} +\end{aligned} +$$ diff --git a/lessons/matematika/1-exponents/1.md b/lessons/matematika/1-exponents/1.md index b8224b2..16bf26b 100644 --- a/lessons/matematika/1-exponents/1.md +++ b/lessons/matematika/1-exponents/1.md @@ -7,13 +7,15 @@ # Exponent Operations ## Multiplication -If an exponented number is multiplied by another exponented number, the exponents would be summed together (instead of being multiplied) +If an exponented number is multiplied by another exponented number, the +exponents would be summed together (instead of being multiplied) \\( a^x \times b^y = (a \times b)^{x + y} \\) * \\( 10^4 \times 2^2 = (10 \times 2)^{4 + 2} = 20^6 \\) ## Division -If an exponented number is multiplied by another exponented number, the exponents would be subtracted together (instead of being divided) +If an exponented number is multiplied by another exponented number, the +exponents would be subtracted together (instead of being divided) \\( a^x \div b^y = (a \div b)^{x - y} \\) * \\( 10^3 \div 2^4 = (10 \div 2)^{(4 - 2)} = 5^2 = 25 \\) @@ -35,13 +37,36 @@ If a number is raised to the power of **zero**, the result is **one**. * \\( 69.420^0 = 1 \\) ## Negative Exponent -If a number is raised to a negative power, also called as an inverse, it would be equal to... +If a number is raised to a negative power, also called as an inverse, it would +be equal to... \\( a^{-b} = \dfrac{1}{a^b} \\) - ## Fractional Exponent - -# Examples - -## Exponential Equation +**Denominator (y)** root of the **number (a)** to the power of the **Numerator (x)**. + +\\( a^{\frac{x}{y}} = \sqrt[y]{a^x} \\) +* \\( 5^{\frac{1}{2}} = \sqrt[2]{5} \\) +* \\( 7^{\frac{2}{3}} = \sqrt[3]{7^2} \\) + +# Exponential Equation +1. $$ +\begin{aligned} +2^{x + 5} &= 64 \\\\ +2^{x + 5} &= 2^6 &\leftarrow\textnormal{ you can change 64 into an exponented two (}2^x\textnormal{)} \\\\ +x + 5 &= 6 &\textnormal{eliminate the 2s} \\\\ +x &= 6 - 5 \\\\ +&= 1 +\end{aligned} +$$ +1. $$ +\begin{aligned} +16^{x + 2} &= 2^5 \\\\ +(2^4)^{x + 2} &= 2^5 \\\\ +2^{4x + 8} &= 2^5 \\\\ +4x + 8 &= 5 \\\\ +4x &= 5 - 8 \\\\ +4x &= -3 \\\\ +x &= -\dfrac{3}{4} +\end{aligned} +$$ diff --git a/lessons/matematika/1-exponents/2-id.md b/lessons/matematika/1-exponents/2-id.md new file mode 100644 index 0000000..31453a9 --- /dev/null +++ b/lessons/matematika/1-exponents/2-id.md @@ -0,0 +1,177 @@ +# Definisi +Sebuah akar merupakan kebalikan dari pangkat. Sebagai contoh: +* \\( 4^2 = 16; \sqrt{16} = 4 \\) +* \\( 5^2 = 25; \sqrt{25} = 5 \\) +* \\( 3^3 = 27; \sqrt[3]{27} = 3 \\) + +# Menyederhanakan Akar Kuadrat +Kamu dapat menyederhanakan sebuah akar dengan memfaktorkan dengan angka lain +yang rasional. Dengan kata lain, membagi bilangan akar dengan kuadrat sempurna. +Setelah itu, akar kuadrat sempurna itu. + +Sebagai contoh: **Sederhanakan** \\( \sqrt{60} \\)! +\\( \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15} \\) +!---( +#--( +## 💡 Penjelasan terperinci +)--# +1. Kita faktorkan 60. Kita pilih faktor terkecil dengan kuadrat yang sempurna (cont. 4, 9, 16, 25). + Kita akan mendapatkan \\(4 \times 15 \\). Untuk membuktikannya, \\( 4 + \times 15 \\) harus sama dengan 60. Dan itu benar! +2. Lalu, akar kuadrat sempurnanya, dalam contoh ini, 4. + \\( \sqrt{4 \times 15} = 2\sqrt{15} \\) +)---! +Contoh lebih banyak: +* \\( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \\) +* \\( \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \\) + +!---( +#--( +## ❕ Ada akar "sisa"! +)--# +Beberapa akar adalah **bilangan tidak rasional**. Bilangan tersebut adalah +bilangan yang tidak bisa direpresentasikan dengan rasio atau pecahan. + +Kita tidak dapat menyederhanakan akar yang tidak rasional. Jadi, akar itu **sudah +dalam bentuk paling sederhana**. +)---! + +# Operasi Akar +Pertama-tama, beberapa terminologi! +1. Koefisien adalah pengali dari akar. Sebagai contoh: \\( 3\sqrt{2} \\); 3 adalah koefisiennya, dan 2 adalah bilangan akarnya. + +## Addition +Root numbers with the same root can be summed together. + +\\( a\sqrt{c} + b\sqrt{c} = (a + b)\sqrt{c} \\) + +* \\( 3\sqrt{2} + 4\sqrt{2} = (3 + 4)\sqrt{2} = 7\sqrt2 \\) +* \\( \sqrt5 + 3\sqrt5 = (1 + 3)\sqrt5 = 4\sqrt5 \\) + +## Subtraction +Root numbers with the same root can be subtracted together. + +\\( a\sqrt{c} - b\sqrt{c} = (a - b)\sqrt{c} \\) + +* \\( 7\sqrt3 - 4\sqrt3 = (7-4)\sqrt3 = 3\sqrt2 \\) + +## Multiplication +Root numbers can be multiplied by any other root numbers. + +\\( a\sqrt{c} \times b\sqrt{d} = (a \times b)\sqrt{c \times d} \\) + +!---( +#--( +## 💡 Tip +)--# +If you multiply a root number with another root number that has the same root, +you can delete the root symbol, aka it becomes a regular number. +For example: + +\\( \sqrt5 \times \sqrt5 = 5 \\) + +!---( +#--( +### ❓ Why? +)--# +Root is the inverse of exponent. If you multiply two numbers together with the +same value, it is an exponent. If two of them meet together, they will fight +and both of them dies (they will cancel out), which means it will be a regular +ol' number. + +In this example: + +\\( \sqrt5 \times \sqrt5 = \sqrt{5^2} = 5 \\) +* Here, \\( \sqrt{5} \\) was powered to 2, which cancels it out. + * **Why do they cancel out?** We can expand this equation even more. + \\( \sqrt5 \times \sqrt5 = \sqrt{5^2} = \sqrt{25} = 5 \\) + * \\( \sqrt{25} \\) is equal to 5, so the answer is 5. + +--- + +This also works for every other number. + +--- + +### In conclusion of these tips: +* If a root is exponented, they will both cancel out. \\( \sqrt[x]{a^x} = a \\) + * Because of that, \\( \sqrt{a} \times \sqrt{a} = a \\) + +)---! +)---! + +## Division +Roots can be divided with any other roots. + +\\( a\sqrt{c} \div b\sqrt{d} = (a \div b)\sqrt{c \div d} \\) + +* \\( 8\sqrt6 \div 4\sqrt3 = (8 \div 4)\sqrt{6 \div 3} = 2\sqrt2 \\) + +# Rationalising Roots +If a fraction has an **irrational** root as the denominator, we should rationalise it. + +!---( +#--( +## ❓ What? Why? +)--# +Usually in mathemathics, a fraction should have a rational number as the +denominator. If you remember, some roots are **irrational**, like \\( \sqrt2 +\\), \\( \sqrt3 \\), \\( \sqrt 5 \\). They cannot be represented as integers or +a valid fraction. + +You don't have to rationalise a root, but it is widely accepted that the +denominator is a rational number. +)---! + +Let's learn how to rationalise a root. + +## Simple Irrational Fraction +If you have a simple irrational fraction like this: + +\\( \dfrac{1}{\sqrt2} \\) + +You can rationalise it by multiplying it with its own denominator. If you +remember correctly, when you multiply a square root with itself, it will return +its own number, therefore a rational number! + +\\( \dfrac{1}{\sqrt2} \cdot \dfrac{\sqrt2}{\sqrt2} = \sqrt{1\sqrt2}{2} \\) + +The value is still (and must be!) the same, rationalising roots are basically +representing numbers in another way. + +### More Examples +* \\( \dfrac{3}{\sqrt5} \cdot \dfrac{\sqrt5}{\sqrt5} = \dfrac{3\sqrt5}{5} \\) +* \\( \dfrac{2}{3\sqrt6} \cdot \dfrac{\sqrt6}{\sqrt6} = \dfrac{2\sqrt6}{3 \cdot 6} = \dfrac{2\sqrt6}{18} \\) (you don't need to multiply it with \\( 3\sqrt6\\)) + +## A little bit more complex fractions +If you have a fraction like this: + +\\( \dfrac{5}{3 + \sqrt2} \\) + +You can rationalise it by multiplying it with itself, but invert the signs. Like this: + +\\( \dfrac{5}{3 + \sqrt2} \cdot \dfrac{3 - \sqrt2}{3 - \sqrt2} \\) +\\( \dfrac{5 \cdot (3 - \sqrt2)}{(3 + \sqrt2)(3 - \sqrt2)} \\) + +!---( +#--( +### ❓ Looks familiar? +)--# +If you look closely, \\( (3 + \sqrt2)(3 - \sqrt2) \\) is the same pattern as +\\( (a + b)(a - b) \\)! This pattern is one of the **special binominal +products**. We use this a lot in algebra. + +So, remember this pattern: +\\( (a + b)(a - b) = a^2 - b^2 \\) +)---! + +Let's continue. + +\\( \dfrac{5 \cdot (3 - \sqrt2)}{(3 + \sqrt2)(3 - \sqrt2)} = \dfrac{5(3 - \sqrt2)}{3^2 - \sqrt{2^2}} \\) + +And if you remember correctly, \\( \sqrt{a^2} = a \\). So, + +\\( \dfrac{5(3 - \sqrt2)}{3^2 - \sqrt{2^2}} = \dfrac{5(3 - \sqrt2)}{9 - 2}\\) +\\( = \dfrac{5(3 - \sqrt2)}{6}\\) + +And that's how you rationalise complex fractions. diff --git a/lessons/matematika/1-exponents/2.md b/lessons/matematika/1-exponents/2.md new file mode 100644 index 0000000..8a15ee3 --- /dev/null +++ b/lessons/matematika/1-exponents/2.md @@ -0,0 +1,176 @@ +# Definition +A root is an inverse of an exponent. For example: +* \\( 4^2 = 16; \sqrt{16} = 4 \\) +* \\( 5^2 = 25; \sqrt{25} = 5 \\) +* \\( 3^3 = 27; \sqrt[3]{27} = 3 \\) + +# Simplifying Square Roots +You can simplify a root by factoring the root number with another number that +is rational. In other words, divide the root number by the perfect squares. +After that, root the perfect square. + +For example: **Simplify** \\( \sqrt{60} \\)! +\\( \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15} \\) +!---( +#--( +## 💡 Detailed explanation +)--# +1. We factorise 60. We choose the smallest factor with a perfect square (e.g. 4, 9, 16, 25). + We will get \\(4 \times 15 \\). To verify, \\( 4 \times 15 \\) must be equal to 60. And it is! +2. Then, root the perfect square, which in this case, is 4. + \\( \sqrt{4 \times 15} = 2\sqrt{15} \\) +)---! +More examples: +* \\( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \\) +* \\( \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \\) + +!---( +#--( +## ❕ There are some "leftover" roots! +)--# +Some roots are **irrational numbers**. Irrational numbers are numbers that +cannot be represented by a ratio, aka a fraction. + +We cannot simplify the roots that is irrational. So, it is **already in its +simplest form**. +)---! + +# Root Operations +Firstly, some terminology! +1. Coefficient is the multiplier of a root. For example: \\( 3\sqrt{2} \\); 3 is the coefficient here, and 2 is the root. + +## Addition +Root numbers with the same root can be summed together. + +\\( a\sqrt{c} + b\sqrt{c} = (a + b)\sqrt{c} \\) + +* \\( 3\sqrt{2} + 4\sqrt{2} = (3 + 4)\sqrt{2} = 7\sqrt2 \\) +* \\( \sqrt5 + 3\sqrt5 = (1 + 3)\sqrt5 = 4\sqrt5 \\) + +## Subtraction +Root numbers with the same root can be subtracted together. + +\\( a\sqrt{c} - b\sqrt{c} = (a - b)\sqrt{c} \\) + +* \\( 7\sqrt3 - 4\sqrt3 = (7-4)\sqrt3 = 3\sqrt2 \\) + +## Multiplication +Root numbers can be multiplied by any other root numbers. + +\\( a\sqrt{c} \times b\sqrt{d} = (a \times b)\sqrt{c \times d} \\) + +!---( +#--( +## 💡 Tip +)--# +If you multiply a root number with another root number that has the same root, +you can delete the root symbol, aka it becomes a regular number. +For example: + +\\( \sqrt5 \times \sqrt5 = 5 \\) + +!---( +#--( +### ❓ Why? +)--# +Root is the inverse of exponent. If you multiply two numbers together with the +same value, it is an exponent. If two of them meet together, they will fight +and both of them dies (they will cancel out), which means it will be a regular +ol' number. + +In this example: + +\\( \sqrt5 \times \sqrt5 = \sqrt{5^2} = 5 \\) +* Here, \\( \sqrt{5} \\) was powered to 2, which cancels it out. + * **Why do they cancel out?** We can expand this equation even more. + \\( \sqrt5 \times \sqrt5 = \sqrt{5^2} = \sqrt{25} = 5 \\) + * \\( \sqrt{25} \\) is equal to 5, so the answer is 5. + +--- + +This also works for every other number. + +--- + +### In conclusion of these tips: +* If a root is exponented, they will both cancel out. \\( \sqrt[x]{a^x} = a \\) + * Because of that, \\( \sqrt{a} \times \sqrt{a} = a \\) + +)---! +)---! + +## Division +Roots can be divided with any other roots. + +\\( a\sqrt{c} \div b\sqrt{d} = (a \div b)\sqrt{c \div d} \\) + +* \\( 8\sqrt6 \div 4\sqrt3 = (8 \div 4)\sqrt{6 \div 3} = 2\sqrt2 \\) + +# Rationalising Roots +If a fraction has an **irrational** root as the denominator, we should rationalise it. + +!---( +#--( +## ❓ What? Why? +)--# +Usually in mathemathics, a fraction should have a rational number as the +denominator. If you remember, some roots are **irrational**, like \\( \sqrt2 +\\), \\( \sqrt3 \\), \\( \sqrt 5 \\). They cannot be represented as integers or +a valid fraction. + +You don't have to rationalise a root, but it is widely accepted that the +denominator is a rational number. +)---! + +Let's learn how to rationalise a root. + +## Simple Irrational Fraction +If you have a simple irrational fraction like this: + +\\( \dfrac{1}{\sqrt2} \\) + +You can rationalise it by multiplying it with its own denominator. If you +remember correctly, when you multiply a square root with itself, it will return +its own number, therefore a rational number! + +\\( \dfrac{1}{\sqrt2} \cdot \dfrac{\sqrt2}{\sqrt2} = \sqrt{1\sqrt2}{2} \\) + +The value is still (and must be!) the same, rationalising roots are basically +representing numbers in another way. + +### More Examples +* \\( \dfrac{3}{\sqrt5} \cdot \dfrac{\sqrt5}{\sqrt5} = \dfrac{3\sqrt5}{5} \\) +* \\( \dfrac{2}{3\sqrt6} \cdot \dfrac{\sqrt6}{\sqrt6} = \dfrac{2\sqrt6}{3 \cdot 6} = \dfrac{2\sqrt6}{18} \\) (you don't need to multiply it with \\( 3\sqrt6\\)) + +## A little bit more complex fractions +If you have a fraction like this: + +\\( \dfrac{5}{3 + \sqrt2} \\) + +You can rationalise it by multiplying it with itself, but invert the signs. Like this: + +\\( \dfrac{5}{3 + \sqrt2} \cdot \dfrac{3 - \sqrt2}{3 - \sqrt2} \\) +\\( \dfrac{5 \cdot (3 - \sqrt2)}{(3 + \sqrt2)(3 - \sqrt2)} \\) + +!---( +#--( +### ❓ Looks familiar? +)--# +If you look closely, \\( (3 + \sqrt2)(3 - \sqrt2) \\) is the same pattern as +\\( (a + b)(a - b) \\)! This pattern is one of the **special binominal +products**. We use this a lot in algebra. + +So, remember this pattern: +\\( (a + b)(a - b) = a^2 - b^2 \\) +)---! + +Let's continue. + +\\( \dfrac{5 \cdot (3 - \sqrt2)}{(3 + \sqrt2)(3 - \sqrt2)} = \dfrac{5(3 - \sqrt2)}{3^2 - \sqrt{2^2}} \\) + +And if you remember correctly, \\( \sqrt{a^2} = a \\). So, + +\\( \dfrac{5(3 - \sqrt2)}{3^2 - \sqrt{2^2}} = \dfrac{5(3 - \sqrt2)}{9 - 2}\\) +\\( = \dfrac{5(3 - \sqrt2)}{6}\\) + +And that's how you rationalise complex fractions. diff --git a/lessons/matematika/1-exponents/3-id.json b/lessons/matematika/1-exponents/3-id.json new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/lessons/matematika/1-exponents/3-id.json diff --git a/lessons/matematika/1-exponents/3.json b/lessons/matematika/1-exponents/3.json new file mode 100644 index 0000000..29e0ed3 --- /dev/null +++ b/lessons/matematika/1-exponents/3.json @@ -0,0 +1,24 @@ +{ + "randomised": false, + "questionCountForRandomisation": 0, + "questions": [ + { + "question": "Is simpliCity a bad game?", + "attachment": "", + "answers": [ + { + "answer": "Yes", + "correct": true + }, + { + "answer": "Maybe", + "correct": false + }, + { + "answer":"No", + "correct": false + } + ] + } + ] +} |