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, good to see you again. Here are the solutions to the problems: Q1: Simplify the following Surds. a) Simplify (6)/(2sqrt(2)-1). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2sqrt(2)+1. (6)/(2sqrt(2)-1) × 2sqrt(2)+12sqrt(2)+1 Step 2: Multiply the numerators and denominators. 6(2sqrt(2)+1)(2sqrt(2)-1)(2sqrt(2)+1) Step 3: Expand the numerator and use the difference of squares formula (a-b)(a+b)=a^2-b^2 for the denominator. 12sqrt(2)+6(2sqrt(2))^2 - 1^2 Step 4: Simplify the denominator. 12sqrt(2)+64 × 2 - 1 = 12sqrt(2)+68 - 1 = 12sqrt(2)+67 The simplified expression is: 12sqrt(2)+67 b) Simplify 2sqrt(3)+22sqrt(3)-2. Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2sqrt(3)+2. 2sqrt(3)+22sqrt(3)-2 × 2sqrt(3)+22sqrt(3)+2 Step 2: Multiply the numerators and denominators. (2sqrt(3)+2)(2sqrt(3)+2)(2sqrt(3)-2)(2sqrt(3)+2) Step 3: Expand the numerator using (a+b)^2=a^2+2ab+b^2 and the denominator using (a-b)(a+b)=a^2-b^2. (2sqrt(3))^2 + 2(2sqrt(3))(2) + 2^2(2sqrt(3))^2 - 2^2 Step 4: Simplify the terms. 4 × 3 + 8sqrt(3) + 44 × 3 - 4 = 12 + 8sqrt(3) + 412 - 4 Step 5: Combine like terms. 16 + 8sqrt(3)8 Step 6: Factor out the common term from the numerator and simplify. 8(2 + sqrt(3))8 = 2 + sqrt(3) The simplified expression is: 2+sqrt(3) Q2: Simplify the following Logarithms. a) Solve _4 (2x+3) - 3 = _4 x. Step 1: Rearrange the equation to group the logarithm terms on one side. _4 (2x+3) - _4 x = 3 Step 2: Use the logarithm property _b A - _b B = _b ((A)/(B)). _4 ((2x+3)/(x)) = 3 Step 3: Convert the logarithmic equation to an exponential equation. If _b A = C, then A = b^C. (2x+3)/(x) = 4^3 Step 4: Calculate 4^3. (2x+3)/(x) = 64 Step 5: Multiply both sides by x. 2x+3 = 64x Step 6: Subtract 2x from both sides. 3 = 64x - 2x 3 = 62x Step 7: Solve for x. x = (3)/(62) The solution for x is: x = (3)/(62) b) If _2 y = 3 - _2 4^3/2, find the value of y. Step 1: Simplify the term 4^3/2. 4^3/2 = (sqrt(4))^3 = 2^3 = 8 Step 2: Substitute this value back into the equation. _2 y = 3 - _2 8 Step 3: Evaluate _2 8. Since 2^3 = 8, _2 8 = 3. _2 y = 3 - 3 _2 y = 0 Step 4: Convert the logarithmic equation to an exponential equation. y = 2^0 Step 5: Calculate 2^0. y = 1 The value of y is: y=1 Send me the next one 📸