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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Here are the solutions to the questions: Q1. Solve the following: i) State the five (5) Laws of Indices 1. Product Rule: a^m × a^n = a^m+n 2. Quotient Rule: (a^m)/(a^n) = a^m-n 3. Power Rule: (a^m)^n = a^mn 4. Zero Exponent Rule: a^0 = 1 (where a ≠ 0) 5. Negative Exponent Rule: a^-n = (1)/(a^n) (where a ≠ 0) ii) Simplify 2a + 5a × 3a - a Step 1: Perform the multiplication first. 5a × 3a = 15a^2 Step 2: Substitute this back into the expression. 2a + 15a^2 - a Step 3: Combine the like terms (2a and -a). 15a^2 + (2a - a) = 15a^2 + a The simplified expression is 15a^2 + a. Q2. Simplify the following: i) (a^3 b^2 c^4)/(abc^-2) and evaluate when a=3, b=(1)/(8), c=2 Step 1: Simplify the expression using the laws of indices. For a: (a^3)/(a^1) = a^3-1 = a^2 For b: (b^2)/(b^1) = b^2-1 = b^1 = b For c: (c^4)/(c^-2) = c^4 - (-2) = c^4+2 = c^6 The simplified expression is a^2 b c^6. Step 2: Substitute the given values a=3, b=(1)/(8), c=2 into the simplified expression. (3)^2 × ((1)/(8)) × (2)^6 Step 3: Calculate the powers. 3^2 = 9 2^6 = 64 Step 4: Perform the multiplication. 9 × (1)/(8) × 64 9 × (64)/(8) 9 × 8 72 The evaluated value is 72. Q3. A deposit of N50 is made on the first day of each month in a saving account that pays 6% compounded monthly. What is the balance at the end of 2 years? This is an annuity due problem. Monthly Payment (PMT) = N50 Annual Interest Rate (r) = 6% = 0.06 Number of compounding periods per year (n) = 12 (monthly) Time (t) = 2 years Total number of payments (N) = n × t = 12 × 2 = 24 Monthly Interest Rate (i) = (r)/(n) = (0.06)/(12) = 0.005 The formula for the future value of an annuity due is: FV_due = PMT × (((1+i)^N - 1))/(i) × (1+i) Step 1: Calculate (1+i)^N. 1+i = 1 + 0.005 = 1.005 (1.005)^24 ≈ 1.127159776 Step 2: Substitute the values into the formula. FV_due = 50 × ((1.127159776 - 1))/(0.005) × (1.005) FV_due = 50 × (0.127159776)/(0.005) × 1.005 FV_due = 50 × 25.4319552 × 1.005 FV_due = 1271.59776 × 1.005 FV_due ≈ 1277.9557 Step 3: Round to two decimal places for currency. The balance at the end of 2 years is N1277.96. Q4. A painter is paid N4.20 per hour for a basic 36 hour week, and overtime is paid at one and a third times this rate. Determine how many hours the painter has to work in a week to earn N212.80. Step 1: Calculate the earnings for the basic 36 hours. Basic hourly rate = N4.20 Basic earnings = 36 hours × N4.20/hour = N151.20 Step 2: Calculate the remaining amount the painter needs to earn. Total desired earnings = N212.80 Remaining earnings = N212.80 - N151.20 = N61.60 Step 3: Calculate the overtime hourly rate. Overtime rate factor = 1 (1)/(3) = (4)/(3) Overtime hourly rate = (4)/(3) × N4.20 = N5.60/hour Step 4: Calculate the number of overtime hours required to earn the remaining amount. Overtime hours = Remaining earningsOvertime hourly rate = N61.60N5.60/hour = 11 hours Step 5: Calculate the total hours the painter has to work. Total hours = Basic hours + Overtime hours = 36 hours + 11 hours = 47 hours The painter has to work a total of 47 hours.