solve the equation (1)/(x)x sqrt(x2 x5)((1)/(sqrt(x)))3 + (2)/(sqrt(x))x [2sqrt(x3)-((1)/(sqrt(x)))^-3] + x(x3sqrt(x))^-(2)/(7) = 4.

Step 1: Simplify the first term of the equation. The first term is (1)/(x) × sqrt(x^2 x^5)((1)/(sqrt(x)))^3. First, convert all terms to powers of x: sqrt(x^5) = x^(5)/(2) (1)/(sqrt(x)) = x^-(1)/(2) ((1)/(sqrt(x)))^3 = (x^-(1)/(2))^3 = x^-(3)/(2) Substitute these into the expression under the square root: (x^2 · x^5)/(2)x^-(3)/(2) = (x^2+5)/(2)x^-(3)/(2) = (x^4)/(2)+(5)/(2)x^-(3)/(2) = (x^9)/(2)x^-(3)/(2) Now, apply the division rule for exponents (a^m/a^n = a^m-n): x^(9)/(2) - (-(3)/(2)) = x^(9)/(2) + (3)/(2) = x^(12)/(2) = x^6 So, the square root becomes sqrt(x^6) = (x^6)^(1)/(2) = x^3. Finally, multiply by (1)/(x): (1)/(x) × x^3 = x^-1 × x^3 = x^3-1 = x^2 The first term simplifies to x^2. Step 2: Simplify the second term of the equation. The second term is (2)/(sqrt(x)) × [2sqrt(x^3) - ((1)/(sqrt(x)))^-3]. Convert terms to powers of x: (2)/(sqrt(x)) = 2x^-(1)/(2) sqrt(x^3) = x^(3)/(2) ((1)/(sqrt(x)))^-3 = (x^-(1)/(2))^-3 = x^(3)/(2) Substitute these into the expression: 2x^-(1)/(2) × [2x^(3)/(2) - x^(3)/(2)] Simplify the expression inside the brackets: 2x^(3)/(2) - x^(3)/(2) = (2-1)x^(3)/(2) = x^(3)/(2) Now, multiply: 2x^-(1)/(2) × x^(3)/(2) = 2x^-(1)/(2)+(3)/(2) = 2x^(2)/(2) = 2x^1 = 2x The second term simplifies to 2x. Step 3: Simplify the third term of the equation. The third term is x(x^3sqrt(x))^-(2)/(7). First, simplify the term inside the parenthesis: x^3sqrt(x) = x^3 · x^(1)/(2) = x^3+(1)/(2) = x^(6)/(2)+(1)/(2) = x^(7)/(2) Now, raise this to the power of -(2)/(7): (x^(7)/(2))^-(2)/(7) = x^(7)/(2) × (-(2)/(7)) = x^-1 Finally, multiply by x: x · x^-1 = x^1-1 = x^0 = 1 The third term simplifies to 1. Step 4: Substitute the simplified terms back into the original equation and solve for x. The original equation was T_1 + T_2 + T_3 = 4. Substituting the simplified terms: x^2 + 2x + 1 = 4 Rearrange the equation into a standard quadratic form: x^2 + 2x + 1 - 4 = 0 x^2 + 2x - 3 = 0 Factor the quadratic equation: (x+3)(x-1) = 0 This gives two possible solutions: x+3 = 0 x = -3 x-1 = 0 x = 1 Step 5: Check the validity of the solutions. For the terms involving sqrt(x) in the original equation to be defined in real numbers, x must be greater than 0. If x = -3, terms like sqrt(x) (e.g., sqrt(-3)) are not real numbers. Therefore, x=-3 is not a valid solution. If x = 1, all terms are well-defined. Let's verify x=1: 1^2 + 2(1) + 1 = 1 + 2 + 1 = 4. This matches the right side of the equation. Thus, the only valid solution is x=1. x=1 Last free one today — make it count tomorrow, or type /upgrade for unlimited.