1. **Index Notation** Index notation is a way to express repeated multiplication of the same number using exponents. For example, $a^n$ means multiplying $a$ by itself $n$ times: $$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$$ 2. **Prime Factorization** Prime factorization is expressing a number as a product of prime numbers. For example, 60 can be factorized as: $$60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$$ 3. **Square Root** The square root of a number $x$ is a number $y$ such that $y^2 = x$. It is denoted as $\sqrt{x}$. For example, $\sqrt{25} = 5$ because $5^2 = 25$. 4. **Finding Square Root by Prime Factorization** To find the square root of a number using prime factorization: - Factorize the number into primes. - Pair the prime factors. - Take one factor from each pair and multiply them. Example: Find $\sqrt{144}$. $$144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2$$ Pairs: $(2^4)$ and $(3^2)$ Take one from each pair: $2^2 \times 3 = 4 \times 3 = 12$ So, $\sqrt{144} = 12$. 5. **Cube Root** The cube root of a number $x$ is a number $y$ such that $y^3 = x$. It is denoted as $\sqrt[3]{x}$. For example, $\sqrt[3]{27} = 3$ because $3^3 = 27$. 6. **Finding Cube Root by Prime Factorization** To find the cube root: - Prime factorize the number. - Group the prime factors in triples. - Take one factor from each triple and multiply. Example: Find $\sqrt[3]{216}$. $$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3$$ Triples: $(2^3)$ and $(3^3)$ Take one from each triple: $2 \times 3 = 6$ So, $\sqrt[3]{216} = 6$. 7. **Highest Common Factor (HCF)** HCF of two or more numbers is the greatest number that divides all of them exactly. To find HCF using prime factorization: - Factorize each number. - Identify common prime factors with the smallest powers. - Multiply these common factors. Example: HCF of 24 and 36. $$24 = 2^3 \times 3$$ $$36 = 2^2 \times 3^2$$ Common factors: $2^2$ and $3^1$ HCF = $2^2 \times 3 = 4 \times 3 = 12$ 8. **Lowest Common Multiple (LCM)** LCM of two or more numbers is the smallest number divisible by all. To find LCM using prime factorization: - Factorize each number. - Take all prime factors with the highest powers. - Multiply them. Example: LCM of 24 and 36. $$24 = 2^3 \times 3$$ $$36 = 2^2 \times 3^2$$ Take highest powers: $2^3$ and $3^2$ LCM = $8 \times 9 = 72$ 9. **Prime Factorization of Large Numbers** Use factor trees or division by primes starting from smallest primes (2, 3, 5, 7, ...). Break down the number step-by-step until all factors are prime. 10. **Using Factor Trees for Multiple-Step Numbers** A factor tree breaks a number into factors repeatedly until all are prime. Example: For 180, 180 / \ 18 10 / \ / \ 2 9 2 5 / \ 3 3 Prime factors: $2^2 \times 3^2 \times 5$ 11. **Expressing Numbers as Powers of Primes** After prime factorization, write the number as product of primes raised to powers. Example: $360 = 2^3 \times 3^2 \times 5$ 12. **HCF and LCM of 3 or More Numbers Using Prime Factors** - For HCF: take common prime factors with smallest powers. - For LCM: take all prime factors with highest powers. Example: 12, 18, 30 $$12 = 2^2 \times 3$$ $$18 = 2 \times 3^2$$ $$30 = 2 \times 3 \times 5$$ HCF = $2^1 \times 3^1 = 6$ LCM = $2^2 \times 3^2 \times 5 = 180$ 13. **Relationship Between HCF × LCM and the Original Numbers** For two numbers $a$ and $b$: $$HCF(a,b) \times LCM(a,b) = a \times b$$ This helps find missing values if one is unknown. 14. **Finding Missing Numbers Using HCF and LCM** If HCF and LCM of two numbers are known, and one number is known, the other can be found by: $$\text{Other number} = \frac{HCF \times LCM}{\text{Known number}}$$ 15. **Determining If a Number Is Perfect Square or Cube from Prime Factorization** - Perfect square: all prime exponents are even. - Perfect cube: all prime exponents are multiples of 3. Example: $36 = 2^2 \times 3^2$ is a perfect square. 16. **Simplifying Fractions Using Prime Factors** - Factor numerator and denominator. - Cancel common prime factors. Example: Simplify $\frac{24}{36}$. $$24 = 2^3 \times 3$$ $$36 = 2^2 \times 3^2$$ Cancel $2^2$ and $3$: $$\frac{2^{3-2} \times 3^{1-1}}{1} = \frac{2}{1} = 2$$ 17. **Solving Word Problems Involving HCF and LCM** Use HCF for problems about grouping or dividing into smaller equal parts. Use LCM for problems about synchronization or common occurrences. 18. **Prime Factorization in Divisibility Problems** Prime factors help check divisibility by seeing if the divisor's prime factors are contained in the number. 19. **Using Prime Factors to Compare Large Numbers** Prime factorization breaks numbers into components making it easier to compare size or divisibility. 20. **Application of Prime Factorization in Problem Solving** Used in arranging objects, grouping items, or finding patterns by understanding the building blocks of numbers. 21. **Mental Estimation of Square Roots and Cube Roots** Use nearby perfect squares or cubes to estimate roots mentally. Example: $\sqrt{50}$ is close to $\sqrt{49} = 7$, so about 7.07. This completes the explanation of all topics in Chapter 1.