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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Hey Roger, good to see you again. Here are the solutions to the exercises: Exercise 1: a) Solve for x: _3(2x + 1) = 3 Step 1: Convert the logarithmic equation to exponential form. The definition of a logarithm states that if _b y = x, then b^x = y. 3^3 = 2x + 1 Step 2: Calculate 3^3 and solve for x. 27 = 2x + 1 27 - 1 = 2x 26 = 2x x = (26)/(2) x = 13 Since 2(13) + 1 = 27 > 0, the solution is valid. x = 13 b) Solve for x: _3(x + 5) - _3(x - 1) = 3 Step 1: Use the logarithm property _b M - _b N = _b ((M)/(N)) to combine the terms. _3((x + 5)/(x - 1)) = 3 Step 2: Convert the logarithmic equation to exponential form. 3^3 = (x + 5)/(x - 1) 27 = (x + 5)/(x - 1) Step 3: Multiply both sides by (x - 1) and solve for x. 27(x - 1) = x + 5 27x - 27 = x + 5 27x - x = 5 + 27 26x = 32 x = (32)/(26) x = (16)/(13) Check for validity: x+5 = (16)/(13) + 5 = (16+65)/(13) = (81)/(13) > 0 x-1 = (16)/(13) - 1 = (16-13)/(13) = (3)/(13) > 0 Both arguments are positive, so the solution is valid. x = (16)/(13) c) Solve for x: 3^3x = (1)/(729) Step 1: Express 729 as a power of 3. 3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243 3^6 = 729 So, 729 = 3^6. Step 2: Rewrite the equation using the power of 3. 3^3x = (1)/(3^6) Step 3: Use the property (1)/(a^n) = a^-n to rewrite the right side. 3^3x = 3^-6 Step 4: Equate the exponents since the bases are the same. 3x = -6 x = (-6)/(3) x = -2 x = -2 Exercise 2: i) Express the following expressions in their simplest form: a) sqrt(12) Step 1: Find the largest perfect square factor of 12. The factors of 12 are 1, 2, 3, 4, 6, 12. The largest perfect square factor is 4. sqrt(12) = sqrt(4 × 3) Step 2: Use the property sqrt(ab) = sqrt(a) × sqrt(b) to simplify. sqrt(12) = sqrt(4) × sqrt(3) sqrt(12) = 2sqrt(3) 2sqrt(3) b) sqrt(50) Step 1: Find the largest perfect square factor of 50. The factors of 50 are 1, 2, 5, 10, 25, 50. The largest perfect square factor is 25. sqrt(50) = sqrt(25 × 2) Step 2: Use the property sqrt(ab) = sqrt(a) × sqrt(b) to simplify. sqrt(50) = sqrt(25) × sqrt(2) sqrt(50) = 5sqrt(2) 5sqrt(2) ii) Rationalize the following surds: a) (2)/(1+sqrt(3)) Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 1+sqrt(3) is 1-sqrt(3). (2)/(1+sqrt(3)) × 1-sqrt(3)1-sqrt(3) Step 2: Expand the numerator and denominator. Use the difference of squares formula (a+b)(a-b) = a^2 - b^2 for the denominator. 2(1-sqrt(3))1^2 - (sqrt(3))^2 2 - 2sqrt(3)1 - 3 2 - 2sqrt(3)-2 Step 3: Simplify the expression. 2(1 - sqrt(3))-2 -(1 - sqrt(3)) -1 + sqrt(3) sqrt(3) - 1 b) (1)/(sqrt(5)-3) Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of sqrt(5)-sqrt(3) is sqrt(5)+sqrt(3). (1)/(sqrt(5)-3) × sqrt(5)+sqrt(3)sqrt(5)+sqrt(3) Step 2: Expand the numerator and denominator. Use the difference of squares formula (a-b)(a+b) = a^2 - b^2 for the denominator. 1(sqrt(5)+sqrt(3))(sqrt(5))^2 - (sqrt(3))^2 sqrt(5)+sqrt(3)5 - 3 sqrt(5)+sqrt(3)2 sqrt(5)+sqrt(3)2 Drop the next question!