This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — here are the solutions to the math questions. 1.2.3 Complete the rule in symbols: a^m ÷ a^n = a^m-n (a^m)^n = a^m × n 1.3 Complete the following table by using your calculator. The first line has been done for you as an example. | a^m | a^n | (a^m)^n | Prime Factors | a^m × n | | :------ | :---- | :---------- | :------------ | :--------------- | | 2^2 | 3 | (2^2)^3 | 64 = 2^6 | 2^2 × 3 = 2^6 | | 3^2 | 4 | (3^2)^4 | 3^8 | 3^2 × 4 = 3^8 | | 3^3 | 2 | (3^3)^2 | 3^6 | 3^3 × 2 = 3^6 | | 2^4 | 5 | (2^4)^5 | 2^20 | 2^4 × 5 = 2^20 | | 5^3 | 2 | (5^3)^2 | 5^6 | 5^3 × 2 = 5^6 | | a^2 | x | (a^2)^x | a^2x | a^2 × x = a^2x | | 2^x | y | (2^x)^y | 2^xy | 2^x × y = 2^xy | 1.3.2 Explain the general rule of what is happening in the table above in your own words. When a power is raised to another power, the exponents are multiplied together, while the base remains the same. 1.3.3 Complete the rule in symbols: (a^m)^n = a^mn 1.3.4 Does this rule apply for every number in the bracket? (2 × 3)^2 × (3 × 2)^2? Show your workings. Yes, the rule (a^m)^n = a^mn applies to each factor within the bracket. Step 1: Rewrite the expression using the rule (ab)^n = a^n b^n. (2 × 3)^2 × (3 × 2)^2 = (2^1 × 3^1)^2 × (3^1 × 2^1)^2 Step 2: Apply the rule (a^m)^n = a^mn to each factor. = (2^1 × 2 × 3^1 × 2) × (3^1 × 2 × 2^1 × 2) Step 3: Simplify the exponents and calculate the value. = (2^2 × 3^2) × (3^2 × 2^2) = (4 × 9) × (9 × 4) = 36 × 36 = 1296 Answer: ✅ Yes, the rule applies to each factor within the bracket. 1.3.5 Is (2 × 3)^2 × (3 × 2)^2 the same as (2 × 3 × 3 × 2)^2? Show your workings. Step 1: Calculate the value of the left-hand side (LHS). (2 × 3)^2 × (3 × 2)^2 = (6)^2 × (6)^2 = 36 × 36 = 1296 Step 2: Calculate the value of the right-hand side (RHS). (2 × 3 × 3 × 2)^2 = (6 × 6)^2 = (36)^2 = 1296 Step 3: Compare the LHS and RHS. Since 1296 = 1296, the expressions are the same. Answer: ✅ Yes, they are the same. ACTIVITY 2: FUNCTIONS AND RELATIONS 2.1 Use the following rule: To find the y-values, add 3 to the x-values and then divide by 5. The rule can be written as y = (x+3)/(5). 2.1.1 Complete the table of values. | x | $y