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.
ScanSolve AI Answer
Right Elijah, let's go. Problem 1: Rationalize the denominator and simplify Step 1: Recall the value of 60^. 60^ = sqrt(3) Step 2: Substitute the value into the expression. sqrt(3) 60^ - 1 = sqrt(3)sqrt(3) - 1 Step 3: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(3) + 1. sqrt(3)sqrt(3) - 1 × sqrt(3) + 1sqrt(3) + 1 Step 4: Perform the multiplication. sqrt(3)(sqrt(3) + 1)(sqrt(3) - 1)(sqrt(3) + 1) = sqrt(3) · sqrt(3) + sqrt(3) · 1(sqrt(3))^2 - 1^2 = 3 + sqrt(3)3 - 1 = 3 + sqrt(3)2 The simplified expression is 3 + sqrt(3)2. Problem 2: Solve for x in 3 _3 (x+4) = _3 24 Step 1: Use the logarithm property n _b M = _b M^n on the left side. _3 (x+4)^3 = _3 24 Step 2: Since the bases of the logarithms are the same, equate the arguments. (x+4)^3 = 24 Step 3: Take the cube root of both sides. x+4 = [3]24 x+4 = [3]8 × 3 x+4 = 2[3]3 Step 4: Solve for x. x = 2[3]3 - 4 Step 5: Check the domain restriction for the logarithm. The argument (x+4) must be positive. Since 2[3]3 is approximately 2 × 1.44 = 2.88, x+4 = 2[3]3 ≈ 2.88, which is positive. So the solution is valid. The value of x is 2[3]3 - 4. Problem 3: Matrix transformation Step 1: A transformation maps a triangle onto a straight line if the determinant of the transformation matrix is zero. The given transformation matrix is M = x-1 & x \\ 1 & 2x . Step 2: Calculate the determinant of the matrix. (M) = (x-1)(2x) - (x)(1) = 2x^2 - 2x - x = 2x^2 - 3x Step 3: Set the determinant to zero and solve for x. 2x^2 - 3x = 0 Factor out x: x(2x - 3) = 0 Step 4: Find the possible values of x. x = 0 or 2x - 3 = 0 2x = 3 x = (3)/(2) The possible values of x are 0 or (3)/(2). Problem 4: Binomial expansion a) Expand (2 + (1)/(4)x)^6 Step 1: Use the binomial theorem (a+b)^n = _k=0^n nk a^n-k b^k with a=2, b=(1)/(4)x, and n=6. (2 + (1)/(4)x)^6 = 60 (2)^6 ((1)/(4)x)^0 + 61 (2)^5 ((1)/(4)x)^1 + 62 (2)^4 ((1)/(4)x)^2 + 63 (2)^3 ((1)/(4)x)^3 + 64 (2)^2 ((1)/(4)x)^4 + 65 (2)^1 ((1)/(4)x)^5 + 66 (2)^0 ((1)/(4)x)^6 Step 2: Calculate each term. 60 (2)^6 (1) = 1 · 64 · 1 = 64 61 (2)^5 ((1)/(4)x) = 6 · 32 · (1)/(4)x = 6 · 8x = 48x 62 (2)^4 ((1)/(4)x)^2 = 15 · 16 · (1)/(16)x^2 = 15x^2 63 (2)^3 ((1)/(4)x)^3 = 20 · 8 · (1)/(64)x^3 = 20 · (1)/(8)x^3 = (5)/(2)x^3 64 (2)^2 ((1)/(4)x)^4 = 15 · 4 · (1)/(256)x^4 = 15 · (1)/(64)x^4 = (15)/(64)x^4 65 (2)^1 ((1)/(4)x)^5 = 6 · 2 · (1)/(1024)x^5 = 12 · (1)/(1024)x^5 = (3)/(256)x^5 66 (2)^0 ((1)/(4)x)^6 = 1 · 1 · (1)/(4096)x^6 = (1)/(4096)x^6 Step 3: Combine the terms. (2 + (1)/(4)x)^6 = 64 + 48x + 15x^2 + (5)/(2)x^3 + (15)/(64)x^4 + (3)/(256)x^5 + (1)/(4096)x^6 b) Hence find the value of (2.025)^6 rounded off to 3 decimal places. Step 1: Set 2 + (1)/(4)x equal to 2.025 to find the value of x. 2 + (1)/(4)x = 2.025 (1)/(4)x = 2.025 - 2 (1)/(4)x = 0.025 x = 0.025 × 4 x = 0.1 Step 2: Substitute x=0.1 into the expanded expression. Since x=0.1 is a small value, the higher power terms will become very small. We can calculate a few terms to get the required precision. (2.025)^6 = 64 + 48(0.1) + 15(0.1)^2 + (5)/(2)(0.1)^3 + (15)/(64)(0.1)^4 + = 64 + 4.8 + 15(0.01) + 2.5(0.001) + (15)/(64)(0.0001) + = 64 + 4.8 + 0.15 + 0.0025 + 0.0000234375 + = 68.9525234375 Step 3: Round the result to 3 decimal places. 68.9525234375 ≈ 68.953 The value of (2.025)^6 rounded off to 3 decimal places is 68.953. Send me the next one 📸