Solve the equation (3 k)2 - 3 k0.5 = 2 8.

Step 1: Simplify the right-hand side of the equation. We have _2 8. Since 8 = 2^3, we can write: _2 8 = _2 2^3 = 3 Step 2: Simplify the second term on the left-hand side using the logarithm property _b M^p = p _b M. The term is _3 k^0.5. _3 k^0.5 = 0.5 _3 k Step 3: Substitute the simplified terms back into the original equation. The original equation is (_3 k)^2 - _3 k^0.5 = _2 8. Substituting the simplified terms, we get: (_3 k)^2 - 0.5 _3 k = 3 Step 4: Let x = _3 k to transform the equation into a quadratic form. x^2 - 0.5x = 3 Step 5: Rearrange the quadratic equation into standard form ax^2 + bx + c = 0. x^2 - 0.5x - 3 = 0 To eliminate the decimal, multiply the entire equation by 2: 2x^2 - x - 6 = 0 Step 6: Solve the quadratic equation for x. We can factor the quadratic equation. We look for two numbers that multiply to 2 × -6 = -12 and add to -1. These numbers are -4 and 3. 2x^2 - 4x + 3x - 6 = 0 Factor by grouping: 2x(x - 2) + 3(x - 2) = 0 (2x + 3)(x - 2) = 0 This gives two possible values for x: 2x + 3 = 0 2x = -3 x = -(3)/(2) x - 2 = 0 x = 2 Step 7: Substitute back x = _3 k and solve for k. Case 1: x = 2 _3 k = 2 Using the definition of logarithm (b^y = x _b x = y): k = 3^2 k = 9 Case 2: x = -(3)/(2) _3 k = -(3)/(2) Using the definition of logarithm: k = 3^-(3)/(2) k = (1)/(3^3)2 k = (1)/(sqrt(3^3)) k = (1)/(sqrt(27)) To simplify, we can write sqrt(27) = sqrt(9 × 3) = 3sqrt(3): k = (1)/(3sqrt(3)) Rationalize the denominator by multiplying the numerator and denominator by sqrt(3): k = (1)/(3sqrt(3)) × sqrt(3)sqrt(3) = sqrt(3)3 × 3 = sqrt(3)9 Both solutions for k are positive, which is required for the logarithm _3 k to be defined. The values for k are 9, sqrt(3)9. Send me the next one 📸